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The 702 statements of Algebraic Topology, by chapter and §. Pick a kind, filter by a word, or put the most cited first.

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I. Coverings

§ 1. Fibred products and cartesian squares

Definition 1 A topological B-space (or simply a B-space) is a topological space X, endowed with a continuous mapping p of X into B. The mapping p is called the projection of the B-space X. 01L9
Definition 2 The topological space X×_BX' is called the fibred product of X and X' over B. The B-space obtained by endowing X×_BX' with the mapping q is called the product B-space of X and X'. 01LA
Example 1 Let X and X' be B-spaces; then the mapping (x, x')↦ (x', x) defines a B-isomorphism of X×_BX' onto X'×_BX. 01LB
Example 2 Let X, X',X'' be B-spaces and let p,p',p'' be their respective projections. The product B-space (X×_BX')×_BX'' is the topological subspace X×_BX'×_BX'' of X× X'× X'' formed by the… 01LC
Example 3 Let X be a B-space and p its projection. The fibred product X×_BX of X and X over B is called the fibred square of X. It is the subspace of X× X formed by the pairs (x, x') such th… 01LD
Example 4 Let (Bᵢ)_i∈ I be a family of topological spaces and let, for each i∈ I, (Xᵢ, pᵢ) and (Yᵢ, qᵢ) be Bᵢ-spaces. Put B =∏_i∈ IBᵢ, X =∏_i∈ IXᵢ and Y =∏_i∈ IYᵢ. Endowed with the continuou… 01LE
Example 5 Let (Xᵢ)_i∈ I and (Yⱼ)_j∈ J be families of B-spaces. Let X and Y be their sums. For each (i, j)∈ I× J, the mapping (x, y)↦ ((i, x),(j, y)) is a B-isomorphism of Xᵢ×_BYⱼ onto the su… 01LF
Definition 3 The topological space ∏_BXᵢ is called the fibred product of the family (Xᵢ)_i∈ I over B. The B-space obtained by endowing ∏_BXᵢ with the mapping p is called the product B-space of… 01LG
Example 1 When the set I is empty, the set ∏_i∈ IXᵢ consists of a single element and the B-space ∏_BXᵢ is identified with B (endowed with the projection Id_B). 01LH
Example 2 When I is not empty, one deduces from the mapping (b, x)↦ x of B× ∏_i∈ IXᵢ into ∏_i∈ IXᵢ, by passing to subspaces, a homeomorphism of ∏_BXᵢ onto the subspace of ∏_i∈ IXᵢ formed by… 01LI
Example 3 When the set I is a set with one element α (resp. with two elements α and β; resp. with three elements α ,β ,γ ), the mapping pr_α (resp. (pr_α, pr_β) ; resp. (pr_α, pr_β, pr_γ)) o… 01LJ
Definition 4 One says that square (1) is a Cartesian square of topological spaces (or, simply, that it is Cartesian) if it is commutative and if, for every topological space Y and every pair of… 01LK
Proposition 1 For square (1) to be Cartesian, it is necessary and sufficient that it be commutative and that, for every B-space Y and every pair of B-morphisms u: Y→ B',v: Y→ X, there exist a un… 01LL
Proposition 2 Let B, B' and X be topological spaces and let p: X→ B,f: B'→ B be continuous mappings. 01LM
Proposition 3 Let 01LN
Proposition 4 Let 01LO
Corollary Let 01LP
Example Let (X, p) be a B-space and A a subspace of B. The square 01LQ
Proposition 5 Let I be a set and, for every i∈ I, let 01LR
Corollary 1 Let X be a B-space, let p be its projection, and let F be a topological space. The square 01LS
Corollary 2 If the square (9) is cartesian for every i∈ I, the square (9') is cartesian. 01LT
Example 1 Let 01LU
Proposition 6 Let I be a set and, for every i∈ I, let 01LV
Example 2 Let (X, p) be a B-space and let (Aₖ)_k∈ K be a family of subspaces of B. Let A be the sum space of the family (Aₖ)_k∈ K and let Y be the sum space of the family (overset-1p(Aₖ))_k∈… 01LW
Proposition 7 Let 01LX
Remark 1 It may happen that squares (12) and (14) are cartesian without square (13) being so (I, p. 139, exerc. 2). 01LY
Remark 2 Let p: X→ B and f: B'→ B be continuous mappings. The mapping g: B'→ B'× B defined by g(b') = (b', f(b')) is a homeomorphism of B' onto the graph G of the mapping f (TG, I, p. 25, c… 01LZ
Remark 3 Let 01M0
Proposition 8 Let 01M1
Remark 1 With the notation of prop. 8, if p is a closed mapping, this is not necessarily so for p'(cf. TG, I, p. 72, example). However, if the mapping p is closed and if A is a subspace of… 01M2
Definition 5 Let X and Y be topological spaces and f: X→ Y a mapping. Let R be the equivalence relation associated with f, and 01M3
Example 1 The composite of two strict mappings is not necessarily strict. In fact, every continuous mapping f: X→ Y is the composite of the mapping pr₂: X× Y→ Y and of the mapping x↦ (x, f(x… 01M4
Example 2 A continuous mapping which is open, or closed, or which possesses a continuous section, is strict. This follows from prop. 3 of TG, I, p. 32 and from prop. 9 of TG, I, p. 22. 01M5
Example 3 In order that a continuous homomorphism of one topological group into another be a strict morphism (TG, III, p. 16, def. 1), it is necessary and sufficient that it be a strict mapp… 01M6
Proposition 9 Let X, Y and Z be topological spaces. Let f: X→ Y be a continuous surjective mapping and g: Y→ Z a mapping. 01M7
Proposition 10 Let X be a topological space and let R be an equivalence relation in X. Let Y be a locally compact topological space. Let S be the equivalence relation in X× Y which is the product… 01M8
Corollary Let X and Y be topological spaces and let f: X→ Y be a continuous mapping. Let T be a locally compact topological space. If the mapping f is strict, the same is true of the mapping… 01M9
Definition 6 Let p: X→ B be a continuous mapping. One says that the mapping p is universally strict if for every cartesian square 01MA
Corollary A continuous mapping which is open, or proper, or which admits in a neighbourhood of every point a continuous section, is universally strict (prop. 8 and example 2). 01MB
Example Let B be a topological space and let (Aᵢ)_i∈ I be a covering of B; let A denote the sum space of the family (Aᵢ)_i∈ I, and let p: A→ B be the canonical mapping. The mapping p is un… 01MC
Remark A closed mapping is not necessarily universally strict (cf. TG, I, p. 96, exerc. 6). 01MD
Proposition 11 Let 01ME
Corollary 1 Suppose that the mapping f is universally strict and surjective and that the mapping p' is universally strict. Then the mapping p is universally strict. 01MF
Corollary 2 Let B and X be topological spaces and let p: X→ B be a continuous mapping. Let (Aᵢ)_i∈ I be a family of subsets of B which is an open covering of B, or else a locally finite closed… 01MG

§ 2. Étale mappings

Proposition 1 Let X and Y be topological spaces and let f: X→ Y be a continuous mapping. The following properties are equivalent: 01MH
Definition 1 Let X and Y be topological spaces. A continuous mapping f: X→ Y is said to be separated if it satisfies the equivalent conditions of proposition 1. 01MI
Proposition 2 Let X, Y, Z be topological spaces and let f: X→ Y and g: Y→ Z be continuous mappings. 01MJ
Remark 1 An injective continuous mapping is separated. 01MK
Remark 2 For a topological space X to be separated (TG, I, p. 52, def. 1), it is necessary and sufficient that the mapping of X into a space reduced to a point be separated. In this case, e… 01ML
Remark 3 Let f: X→ Y be a continuous and separated mapping. If the space Y is separated, the space X is separated. This follows from remark 2 and proposition 2 applied with a space Z reduce… 01MM
Remark 4 Let f: X→ Y be a continuous and separated mapping, y a point 01MN
Remark 5 Let Y be a topological space. Let (Aᵢ)_i∈ I be a family of subsets of Y and let X be the sum space of the family (Aᵢ)_i∈ I. The canonical mapping of X into Y is separated. 01MO
Proposition 3 Let f: X→ Y be a continuous and separated mapping. Every continuous section of f induces a homeomorphism of Y onto a closed subset of X. 01MP
Proposition 4 Let 01MQ
Proposition 5 Let X, Y, Z be topological spaces and let f: X→ Y,g: Y→ Z be continuous mappings. Suppose the mapping g separated and the mapping g∘ f proper. Then the mapping f is proper. 01MR
Definition 2 Let E and B be topological spaces, let p: E→ B be a mapping and let x be a point of E. One says that the mapping p is topologically etale at x if there exists a neighbourhood U of… 01MS
Remark 1 The set of points of E at which a mapping p: E→ B is etale is an open set U of E, and the restriction of p to U is an etale mapping of U into B. 01MT
Remark 2 An etale mapping is continuous and open (TG, I, p. 33, prop. 5) ; in particular, the image of an etale mapping is open. Conversely, if p: E→ B is a continuous and open mapping, and… 01MU
Example 1 Let B be a topological space and (Uᵢ)_i∈ I a family of subsets of B. Let E denote the sum space of the family (Uᵢ)_i∈ I and p: E→ B the mapping deduced from the canonical injection… 01MV
Example 2 Let U be an open set of C. In order that a holomorphic function f: U→ C be an etale mapping, it is necessary and sufficient that its derivative nowhere vanish. 01MW
Example 3 An etale morphism of varieties (VAR, R, 5.7.8) is a topologically etale mapping, but there exist morphisms of real varieties which are topologically etale and which are not etale m… 01MX
Proposition 6 Let X, Y, Z be topological spaces and let f: X→ Y,g: Y→ Z be mappings. 01MY
Corollary 1 Let B be a topological space. A B-morphism of one étalé B-space into another is étale. 01MZ
Corollary 2 Let B be a topological space; a bijective B-morphism of one étalé B-space into another is a B-isomorphism. 01N0
Corollary 3 Let p: E→ B be an étale mapping. Every continuous section of p induces a homeomorphism of B onto an open subset of E. 01N1
Corollary 4 Let p: E→ B be an étale and separated mapping. Suppose that E is connected and that p admits a section. Then p is a homeomorphism. 01N2
Proposition 7 Let p: E→ B be a continuous mapping. In order that the mapping p be étale, it is necessary and sufficient that it be open and that the diagonal Δ_E of E×_BE be open in E×_BE. 01N3
Proposition 8 Let 01N4
Corollary Let B be a topological space. The fibre product of two étale B-spaces is an étale B-space. 01N5
Remark 3 Let 01N6
Proposition 9 Let p: E→ B be a continuous mapping. In order that the mapping p be étale, it is necessary and sufficient that every point of E possess an open neighbourhood which is the image of… 01N7
Remark 1 Let p: E→ B be a continuous and open mapping. In order that an open set U of E be the image of a continuous section of p over an open set of B, it is necessary and sufficient that… 01N8
Remark 2 Let p: E→ B be a continuous mapping. Suppose that every point of B possesses an open neighbourhood V such that there exists a continuous section of p over V. Such a mapping p is no… 01N9
Proposition 10 Let E, B and Z be topological spaces, p: E→ B an étale mapping and f: Z→ B a continuous mapping. 01NA
Proposition 11 Let E, B and Z be topological spaces, p: E→ B and f: Z→ B continuous mappings. Let g and g' be continuous liftings of f to E. Let W be the set of points of Z where g and g' coincid… 01NB
Corollary 1 If the space Z is connected and if the mapping p is étale and separated, continuous liftings of f to E which coincide at one point are equal. 01NC
Corollary 2 Let p: E→ B be an étale and separated mapping. If the space E is connected, the group Aut_B(E) operates freely on E. 01ND
Corollary 3 Let E, B and Z be topological spaces, let p: E→ B be an étale and separated mapping, and let f: Z→ B be a continuous mapping. For i= 1, 2, let Uᵢ be an open subset ( resp. closed s… 01NE
Proposition 12 Let p: E→ B be an étale mapping. In order that the mapping p be separated, it is necessary and sufficient that for every open subset V of B and every pair (s, s') of continuous sec… 01NF
Theorem 1 Let X, Y, Z be topological spaces, let f: Z→ X× Y be an étale and separated mapping, and let y₀ be a point of Y. Let s: X× Y→ Z be a section of f. Suppose that the restriction of s… 01NG
Lemma Let U be an open set of X, let V be a connected open set of Y, and let (x₁, y₁)∈ U× V. Suppose that the restriction of s to U× y₁ is continuous. Let σ be a continuous section of f… 01NH
Corollary 1 Let X, Y, E, B be topological spaces. Let p: E→ B be an étale and separated mapping, let h: X× Y→ B be a continuous mapping and let y₀ be a point of Y. Let g: X× Y→ E be a mapping… 01NI
Remark If, in theorem 1, it is not supposed that the space Y is locally connected, the conclusion of the theorem is no longer necessarily true (I, p. 141, exerc. 7). 01NJ
Theorem 2 Let B be a topological space and A a subspace of B. Make one of the following hypotheses: 01NK
Lemma 1 Each of properties (i) to (iv) of theorem 2 implies property (PCV). 01NL
Lemma 2 Let B be a topological space, and let K and L be quasi-compact subsets of B. Suppose that for every point a of K and every point b of L, there exist disjoint neighbourhoods of a an… 01NM
Lemma 3 Let B be a topological space and A a subspace of B such that the pair (B,A) satisfies property (PCV) of I, p. 37. Then every continuous section over A of an étale mapping p: E→ B e… 01NN
Theorem 3 Let E and B be topological spaces and let p: E→ B be an étale and separated mapping. Suppose that E is connected and that B is locally connected. Let W be a basis for the topology… 01NO
Lemma 4 Let B be a topological space, and let W and W' be bases for the topology of B. There exists a subset V of W' which is a basis for the topology of B and such that Card(V)leqslant Ca… 01NP
Lemma 5 a) Let c= (U₀, . . . ,Uₙ) and c'= (U'₀, . . . ,U'ₘ) be caterpillars such that U₀= U'₀ and p(c) =p(c'). Then c=c'. 01NQ
Remark From the foregoing, if the topology of B admits a countable basis, the same is true of that of E and the fibres of E are countable (cf. TG, I, p. 88, Poincaré-Volterra theorem). 01NR

§ 3. Sheaves

Definition 1 A presheaf on B, relative to a basis B of the topology of B, is a projective system of sets, relative to the set of indices B ordered by the inclusion relation. 01NS
Definition 2 Let F= (F(U), f_UV) be a presheaf on B. One says that F is a sheaf on B if, for every open subset U of B and every family (Uᵢ)_i∈ I of open subsets of B, with union U, the followin… 01NT
Remark Let F be a presheaf on B. For every open set U of B, f_∅ U is a map of F(U) into F(∅ ), hence F(∅ ) is not empty as soon as there exists an open set U for which F(U) is not empty… 01NU
Definition 3 Let B be a topological space, B a basis of the topology of B,F= (F(U), f_UV) and G= (G(U), g_UV) presheaves on B relative to B. A morphism of presheaves from F to G is called a pro… 01NV
Example 1 Let B be a topological space, let (E, p) and (E', p') be B-spaces and let f: E→ E' be a B-morphism. For every open subset U of B, define the mapping f_U:S(U; E)→ S(U; E') by f_U(s)… 01NW
Example 2 Let B be a topological space, B a basis of the topology of B, F= (F(U), f_UV) a presheaf on B relative to B and L= (L(U)) a subpresheaf of F. For every open set U∈ B, denote by i_U… 01NX
Example 3 Let B be a topological space, B a basis of the topology of B and I a set. For every i∈ I, let Fᵢ= (Fᵢ(U), f_i,UV) be a presheaf on B relative to B. Denote by F the product presheaf… 01NY
Example 4 Let X be a differential manifold of class C^∞ over R and let C^∞(X;R) be the sheaf on X of numerical functions of class C^∞. If P is a differential operator with C^∞ coefficients o… 01NZ
Proposition 1 The mapping p: E_F→ B is etale. For every open set U∈ B and every s∈ F(U), the mapping σ_F(U, s) is therefore a continuous section of p over U. 01O0
Definition 4 The spread B-space (E_F, p) defined above is called the spread B-space associated with the presheaf F. For x∈ B, the fibre of E_F at x is called the stalk of the presheaf F at x an… 01O1
Remark Let F be a sheaf on B relative to the base B. Let B' be an open subset of B, let B' be a basis of the topology of B' such that B'⊂ B. Let F|B' be the presheaf on B' relative to the… 01O2
Example 1 Let B be a topological space, B a base of the topology of B and F a set. Take for F the presheaf on B relative to B defined by F(U) = F for every U∈ B and r_UV = Id_F for every pai… 01O3
Example 2 Let B be a topological space and (E, p) a B-space. Let F be the sheaf on B of continuous sections of (E, p). The mapping (U, s, x)↦ s(x) of X_F into E is compatible with the equiva… 01O4
Example 3 Let B be a topological space, B a base of the topology of B,F a presheaf on B relative to B and L a sub-presheaf of F. Then the set X_L is contained in the set X_F and the equivale… 01O5
Proposition 2 The mapping j_F is the inverse B-isomorphism of e_F. 01O6
Corollary For every a∈ B, the mapping (σ_F)ₐ:Fₐ→ widetildeFₐ is bijective. 01O7
Proposition 3 Let B be a topological space, let B be a basis of the topology of B, let F= (F(U), f_UV) be a presheaf on B relative to B, let widetildeF be the associated sheaf and let σ_F:F→ wid… 01O8
Lemma Let U be an open subset of B and s: U→ E_F a continuous section of E_F over U. For every point a of U, there exist an open set V∈ B such that a∈ V and V⊂ U, and an element v of F(V… 01O9
Corollary 1 Let B be a topological space, F a presheaf on B,widetildeF the associated sheaf and σ_F:F→ widetildeF the canonical morphism. For F to be a sheaf, it is necessary and sufficient th… 01OA
Remark Let B be a topological space, F a presheaf on B, G a sheaf on B and φ :F→ G a morphism of presheaves. The canonical morphism σ_G:G→ widetildeG is an isomorphism by corollary 1. By… 01OB
Corollary 2 Let B be a topological space, F and G sheaves on B and φ a morphism of sheaves from F into G. The following assertions are equivalent: 01OC
Scholium Let B be a topological space. To every sheaf F on B, one associates an étalé B-space E_F (I, p. 50, def. 4). To every étalé B-space T, one associates the sheaf S(T) on B of its con… 01OD
Example 1 Let B be a topological space, A a subspace of B and F= (F(U), f_UV) a presheaf on A. Let i: A→ B denote the canonical injection. Then one has i_*(F) = (F'(U), f'_UV) where for ever… 01OE
Remark The canonical morphism σ_G:G→ widetildeG of I, p. 53 corresponds, in terms of étalé spaces, to the isomorphism j_G of proposition 2 of I, p. 53. It follows that u^*(σ_G) is an isom… 01OF
Example 2 Let B be a topological space and A a subspace of B. Let i: A→ B denote the canonical injection. For every sheaf G on B, let G_A denote the sheaf i^*(G), and say that G_A is the she… 01OG
Remark 1 Let A, B, C be topological spaces, let u: A→ B,v: B→ C be continuous mappings; put w=v∘ u. Let G be a presheaf on C. 01OH
Remark 2 If γ :G₁→ G₂ is a presheaf morphism on B, the presheaf morphisms β_G₂∘ γ and u_*u^*(γ )∘ β_G₁ are equal. In fact, if V is an open set of B and s∈ G₁(V),β_G₁(s) is the section t of… 01OI
Proposition 4 Let A and B be topological spaces, u: A→ B a continuous mapping, G a presheaf on B,F a sheaf on A. 01OJ
Proposition 5 Let A and B be topological spaces and let u: A→ B be a continuous mapping. 01OK
Example 1 Consider a topological space B, a subspace A of B, and denote by i: A→ B the canonical injection. Let (E, p) and (E', p') be B-spaces. Take for G the sheaf Mor_B(E; E') (I, p. 45… 01OL
Example 2 Let A, B, C be topological spaces and let u: A→ B, v: B→ C be continuous mappings; put w=v∘ u. Let E and E' be C-spaces. The canonical morphism from Mor_C(E; E')_A into Mor_A(E_A… 01OM
Definition 5 Let p: E→ B be an étale mapping. One says that the mapping p is soft, or that the étale B-space (E, p) is soft, if every continuous section of p over a closed subspace of B extends… 01ON
Proposition 6 Let B be a topological space, F a sheaf on B and (Aᵢ)_i∈ I a locally finite closed covering of B. In order that the sheaf F be soft, it is necessary and sufficient that, for every… 01OO
Corollary 1 Let B be a paracompact space, F a sheaf on B and (Uᵢ)_i∈ I an open covering of B. If, for every i∈ I, the induced sheaf F|Uᵢ is soft, then the sheaf F is soft. 01OP
Corollary 2 Let B be a paracompact space, F a sheaf on B and (Aᵢ)_i∈ I a locally finite closed covering of B. In order that the sheaf F be soft, it is necessary and sufficient that the followi… 01OQ

§ 4. Coverings

Definition 1 Let B be a topological space. Let E be a B-space and let p be its projection. We say that E is a locally trivial fibred B-space if every point of B possesses a neighbourhood V such… 01OR
Remark 1 Let (E, p) be a locally trivial fibred B-space and let A be a subset of B. The A-space E_A= (overset-1p(A), p_A) deduced from E by passing to the subspaces is a locally trivial fib… 01OS
Remark 2 Let (E, p) be a B-space and let (Eᵢ)_i∈ I be a partition of E consisting of open sets. 01OT
Definition 2 A covering is a locally trivial fibre space all of whose fibres are discrete. 01OU
Proposition 1 Let B and E be topological spaces and let p: E→ B be a mapping. The following conditions are equivalent: 01OV
Corollary 1 Let B and E be topological spaces and let p: E→ B be a mapping. In order that E, endowed with the mapping p, be a covering of B, it is necessary and sufficient that, for every poin… 01OW
Remark Let B be a topological space. Let E be a B-space and let p be its projection. If E is a covering of B, it follows from corollary 1 above that the mapping p is étale (I, p. 28, Def… 01OX
Corollary 2 Let B be a topological space, let E be a B-space and let p be its projection. Suppose that p is étale and separated and that the space B is connected. In order that E be a triviali… 01OY
Proposition 2 Let B and B' be topological spaces, let (E, p) be a B-space and let (E', p') be a B'-space. Suppose that E and E' are locally trivial fibre spaces ( resp. coverings). The space E×… 01OZ
Corollary 1 Let A be a topological space, B, B' A-spaces. Let E and E' be locally trivial fibre spaces ( resp. coverings) with bases B and B'; let p and p' denote their projections. The B×_AB'… 01P0
Corollary 2 Let 01P1
Corollary 3 Let B be a topological space. Let E and E' be locally trivial fibre B-spaces ( resp. coverings). Then E×_BE' is a locally trivial fibre B-space ( resp. a covering of B). It is triv… 01P2
Proposition 3 Let 01P3
Remark If, in proposition 3, one weakens the hypothesis on the mapping f by supposing it only universally strict and surjective, and if p' is a covering, the mapping p is etale (I, p. 31… 01P4
Proposition 4 Every covering of a nonempty connected space has a degree. 01P5
Theorem 1 Let E, B be topological spaces and p: E→ B a mapping. The following conditions are equivalent: 01P6
Lemma Let X and Y be topological spaces. In order that a mapping f: X→ Y be closed, it is necessary and sufficient that for every point y of Y and every neighbourhood W of the fibre over… 01P7
Remark A continuous, separated, open, proper mapping with finite fibres is not necessarily a covering. This is the case for the mapping x↦ x² of R into [0; +∞ [. 01P8
Corollary 1 Let E and B be topological spaces and let p: E→ B be a proper and separated mapping. The set U of points b of B such that p is étale at every point of the fibre E_b is open in B. T… 01P9
Corollary 2 Let B be a separated topological space and let E be a B-space. Suppose that E is compact and that its projection p: E→ B is étale. Then, E is a finite covering of B. 01PA
Corollary 3 Let B be a topological space, let (E, p) and (E', p') be B-spaces and let f: E'→ E be a B-morphism. Suppose that E' is a locally finite covering of B. 01PB
Corollary 4 Let 01PC
Corollary 5 Let B be a connected topological space and let (E, p) be a finite covering of B. The space E has only a finite number of connected components, and each of them is a covering of B. 01PD
Proposition 5 Let B be a topological space, let E and E' be B-spaces, and let f: E'→ E be a B-morphism. We assume that E is a locally finite covering and that the space E', endowed with the mapp… 01PE
Remark If (E, p) is a covering of B and if (E', f) is a covering of E, the mapping p∘ f is étale (I, p. 29, prop. 6, a)) and separated (I, p. 25, prop. 2, a)). But it can happen, see exer… 01PF
Proposition 6 Let B be a topological space and let E be a B-space; let us denote by p its projection. Suppose that the space E is locally connected. In order that E be a covering of B, it is nec… 01PG
Corollary 1 Let B be a locally connected topological space, let (E, p) be a covering of B and let E' be an open and closed subset of E. The B-space (E', p|E') is a covering and p(E') is open a… 01PH
Corollary 2 Let B be a locally connected topological space, let (E, p), (E', p') be coverings of B and let f, g: E'→ E be B-morphisms. For every point b of B, let us denote by f_b, g_b: E_b→ E… 01PI
Corollary 3 Let B be a connected and locally connected topological space, let (E, p) and (E', p') be coverings of B. For every point b of B, the mapping f↦ f_b of C_B(E'; E) into C(E'_b; E_b)… 01PJ
Corollary 4 Let B be a connected and locally connected topological space, let (E, p) be a covering of B, and let b be a point of B. In order that E be a trivializable covering, it is necessary… 01PK
Corollary 5 Let 01PL
Proposition 7 Let B be a topological space, let E and E' be B-spaces, and let f: E'→ E be a B-morphism. It is assumed that E' is a covering of B and that the space B is locally connected. 01PM
Lemma If U is a connected component of E', the restriction of f to U induces a homeomorphism of U onto a connected component of E. 01PN
Remark Let B be a topological space, let (E, p) and (E', p') be B-spaces and let f: E'→ E be a B-morphism. 01PO
Corollary Let B be a connected and locally connected topological space. Let E and E' be coverings of B. Let b be a point of B. Let f: E'→ E be a B-morphism and let f_b: E'_b→ E_b be the mapp… 01PP
Proposition 8 A covering of a paracompact space ( I, p. 69) is a paracompact space. 01PQ
Lemma Let E be a separated topological space. Suppose that E has a locally finite open covering (Vᵢ)_i∈ I such that, for every i∈ I,overlineVᵢ is a paracompact space. Then the space E is… 01PR
Remark One can prove that a covering of a metrisable space is metrisable (I, p. 145, Exer. 1) and that a connected covering of a locally compact countable at infinity space is itself loca… 01PS
Definition 3 A sheaf F on a topological space B is said to be locally constant if every point of B possesses an open neighbourhood U such that the induced sheaf F|U is isomorphic to a constant… 01PT
Proposition 9 In order that a sheaf be locally constant, it is necessary and sufficient that the associated étalé space be a covering. 01PU
Corollary In order that an étalé B-space be a covering, it is necessary and sufficient that the sheaf of its sections be locally constant. 01PV
Proposition 10 If the set I is finite, the B-morphism Φ is an isomorphism. 01PW
Proposition 11 If the set I is finite, or if the space B is locally connected, the canonical morphism ψ : F→ ∏ Fᵢ is an isomorphism. 01PX
Corollary 1 A finite product of locally constant sheaves is locally constant. 01PY
Corollary 2 Let (Fᵢ)_i∈ I be a family of sheaves on B. Suppose that the space B is locally connected and that every point of B possesses an open neighbourhood U such that, for every i∈ I, the… 01PZ
Remark 1 Let (Eᵢ, pᵢ)_i∈ I be a family of coverings of the topological space B. Suppose that the space B is locally connected and that there exists an open covering (Uⱼ)_j∈ J of B such that… 01Q0
Remark 2 With the preceding notations, the canonical B-morphism Φ : E→ ∏_BEᵢ is bijective. The question is in fact of a local nature on B, and one may suppose that, for every i∈ I, the B-sp… 01Q1
Proposition 12 Suppose that the space B is locally connected and that the B-spaces E and E' are coverings. Then the sheaf Mor_B(E; E') is locally constant and, for every b∈ B, the mapping θ_b is… 01Q2
Corollary 1 Let B be a topological space and A a subspace of B. Suppose that the spaces A and B are locally connected. Let E and E' be coverings of B. Then the canonical morphisms ψ :Mor_B(E… 01Q3
Corollary 2 Let B be a topological space and A a subspace of B. Suppose that the spaces A and B are locally connected and that the pair (B,A) has property (PCV) of I, p. 37. Let E and E' be co… 01Q4

§ 5. Principal fibre spaces

Definition 1 Let G be a topological group and B a topological space. A (right) principal fibre space with base B and group G is a B-space (E, p) endowed with a right operation of G on E (A, I… 01Q5
Example 1 Let (E, p) be a principal fibre space with base B and group G, and let A be a subspace of B. The subspace E_A=overset-1p(A) of E is stable under the operation of G, and the mapping… 01Q6
Example 2 Let (E, p) be a principal fibre space with base B and group G; let (E', p') be a principal fibre space with base B' and group G'. We define a law of operation on the right of the g… 01Q7
Example 3 In particular, when B = B', the B-space E×_BE' is identified with the principal fibre space with group G× G' induced by E× E' over the diagonal Δ_B of B× B. Endowed with this struc… 01Q8
Example 4 Let E be a principal fibre space with base B and group G and let F be a topological space. The (B× F)-space E× F endowed with the law of operation (x, y)· g= (x· g, y), for x∈ E,y∈… 01Q9
Example 5 Let B be a topological space and (E, p) a principal fibre space with base B and group G. Let B' be a topological space and f: B'→ B a continuous mapping. The group G operates on th… 01QA
Proposition 1 Every morphism of principal fibre spaces is an isomorphism. 01QB
Corollary Let G be a topological group, (E, p) and (E', p') principal fibre spaces of group G and with bases B and B' respectively. Let f: B'→ B and f': E'→ E be continuous mappings such tha… 01QC
Proposition 2 Let (E, p) be a principal fibre space with base B and group G, and let ε be the section b↦ (b, e) of the trivial principal fibre space (B× G, pr₁), where e denotes the identity ele… 01QD
Corollary A principal fibre space is trivializable if and only if it admits a continuous section. 01QE
Proposition 3 Let E and B be topological spaces, p: E→ B a continuous mapping and G a topological group operating on the right on E. The following conditions are equivalent: 01QF
Remark 1 Let (E, p) be a principal fibre space with base B and group G. With the notation of proposition 3, the mapping θ⁻¹: E×_BE→ E× G is a trivialisation of the principal fibre space pr₁… 01QG
Corollary 1 Let G be a topological group operating continuously on the right in a topological space E. The following conditions are equivalent: 01QH
Corollary 2 Let G be a topological group, let H be a subgroup of G, and let p: G→ G/H be the canonical mapping. If the mapping p possesses a continuous section over a nonempty open set of G/H… 01QI
Remark 2 This situation presents itself in particular when G is a real Lie group, of finite dimension, countable at infinity, operating transitively and analytically on an analytic manifold… 01QJ
Definition 2 Let B be a topological space and let G be a group. Endow G with the discrete topology. A principal fibre space with base B and group G is called a principal covering of B with grou… 01QK
Proposition 4 Let B be a topological space, (E, p) a B-space, and G a discrete topological group operating on the right on E. The following assertions are equivalent: 01QL
Example 1 Let E be a topological space, G a discrete topological group operating continuously and freely in E on the right. In order that the canonical mapping p: E→ E/G make E a principal c… 01QM
Example 2 Let q: E→ X be an étale and separated mapping. The group Aut_X(E), endowed with the discrete topology, operates continuously on the left in E. If the space E is connected, this ope… 01QN
Theorem 1 Let G be a discrete group operating continuously on the right in a topological space E. Suppose that every point of E possesses a neighbourhood U such that U∩ (U· g) =∅ for every e… 01QO
Example 1 Let n be an integer geqslant 0, let Pₙ(R) be the projective space of dimension n (TG, VI, p. 13) and let Sₙ be the unit sphere of Rⁿ⁺¹ (TG, VI, p. 9). The canonical mapping of Sₙ o… 01QP
Example 2 Every covering of degree 2 possesses a unique structure of principal covering of group Z/2Z. 01QQ
Example 3 Let X be a separated differentiable manifold locally of finite dimension over R, and let widetildeX be the manifold of orientations of the tangent bundle of X (VAR, R, 10.2.4). The… 01QR
Corollary 1 Let G be a discrete topological group operating continuously on the right in a topological space E. The following conditions are equivalent: 01QS
Corollary 2 Let G be a topological group and let H be a discrete subgroup of G. Let H operate on G by right translations. Then the canonical mapping of G into the space G/H of left classes mod… 01QT
Example 4 Endowed with the canonical mapping of R onto T=R/Z (TG, V, p. 2),R is a principal covering of R/Z of group Z. 01QU
Remark 1 Let G be a separated topological group, K a compact subgroup of G and H a discrete subgroup of G. The group G operates continuously and properly on the right in the space Kbackslas… 01QV
Corollary 3 Let G and G' be topological groups, and let φ : G→ G' be a continuous, open and surjective homomorphism. Suppose that the kernel H of φ is discrete. Then, for the operation of H in… 01QW
Remark 2 Let G and G' be topological groups and let φ : G→ G' be a continuous open homomorphism. If the group G' is connected, the homomorphism φ is surjective. In fact, φ (G) is an open, t… 01QX
Remark 3 Let G be a locally compact group countable at infinity and let G' be a separated topological group whose underlying space is a Baire space. Every continuous surjective homomorphism… 01QY
Example 5 For every integer n > 0, let us denote by μₙ the group of nth roots of unity in C (A, V, p. 75). The mapping z↦ zⁿ makes C^* into a principal covering of C^* with group μₙ, this gr… 01QZ
Example 6 Endowed with the mapping z↦ e^2π iz (TG, VIII, p. 8, remark), the space C is a principal covering of C^* with group Z. The mapping x↦ e^2π ix=e(x) of R onto S₁ makes R into a princ… 01R0
Definition 3 Let B be a nonempty topological space. A covering E of B is said to be Galois if it is connected and if, for every point b of B, the operation of the group Aut_B(E) of B-automorphi… 01R1
Proposition 5 Let B be a nonempty topological space and let E be a Galois covering of B. The mapping (h, x)↦ h(x) of Aut_B(E)× E into E is a right operation law of the group Aut_B(E)^∘ on E whic… 01R2
Theorem 2 Let B be a connected topological space, let E be a covering of B, connected and nonempty. The following properties are equivalent: 01R3
Proposition 6 Let B be a topological space and G a group. Let (E, p) be a principal covering of B with group G. Suppose the space B connected and locally connected. Let E₀ be a connected compone… 01R4
Remark If, in proposition 6, one does not suppose the space B locally connected, it may happen that the space E₀ is not a covering of B (I, p. 147, exerc. 1). 01R5
Example 1 Let F be a topological space and let G be a topological group operating continuously on the left in F. If the topological space G is endowed with the operation of G by right transl… 01R6
Example 2 Example 1 is generalized as follows. Let B and F be topological spaces and let G be a topological group operating continuously on the left in F. By passing to the quotients, the ma… 01R7
Example 3 Analogously, let E be a topological space and let G be a topological group operating continuously on the right in E. If the topological space G is endowed with the operation of G b… 01R8
Lemma The mapping λ is a homeomorphism. 01R9
Proposition 7 Let B be a topological space, let G be a topological group, let (E, p) be a principal fibred space over B with group G and let F be a topological space in which the group G operate… 01RA
Example 4 Let B be a topological space, G a topological group, and let (E, p) be a principal fibred B-space with group G. Let H be a subgroup of G. 01RB
Example 5 Let B be a topological space, let G be a topological group, and let E be a principal fibred B-space with group G. Let F be a topological homogeneous space relative to G (TG, III, p… 01RC
Example 6 Let B be a topological space, let G be a topological group, let (E, p) be a principal fibred B-space with group G. Let H be a topological group and let f: G→ H be a morphism of top… 01RD
Definition 4 Let B be a topological space and let G be a topological group. A locally trivial fibre space X over B is said to be associated with a principal fibre bundle E over B with group G i… 01RE
Definition 5 Let B be a topological space, let G be a discrete topological group and let E be a principal covering of B with group G. A covering X of B is said to be associated with the princip… 01RF
Proposition 8 Let B be a connected nonempty topological space. Let G be a discrete group and let (E, p) be a principal covering of B with group G. Suppose that E is connected. With the above not… 01RG
Corollary 1 Let (E, p) be a principal covering of B of group G; suppose that E is connected and nonempty. A covering X of B is associable to E if and only if the covering p^*(X) is trivializab… 01RH
Corollary 2 Let B be a connected and locally connected topological space, let (E, p) be a principal covering of B of group G. Let E₀ be a connected component of E and let G₀ be the subgroup of… 01RI
Proposition 9 Let B be a topological space, let E be a principal covering of B of group G, connected and nonempty. Let F be a nonempty G-set endowed with the discrete topology. In order that the… 01RJ
Proposition 10 Let B be a topological space, let (E, p) be a principal covering of B of group G and let X be a covering of B. Suppose that E and X are connected and nonempty. The following proper… 01RK
Corollary Let E be a Galois covering of B and let X be a covering of B, connected and nonempty. Suppose that X is a finite covering or else that the space B is locally connected. If there ex… 01RL
Theorem 3 Let B be a nonempty, connected and locally connected topological space. Every covering of B is associable to a Galois covering; every finite covering of B is associable to a finite… 01RM
Definition 6 Let B be a topological space, let G be a topological group, and let U= (Uᵢ)_i∈ I be an open covering of B. A continuous 1-cocycle on B with values in G, subordinate to U, is by def… 01RN
Definition 7 Let B be a topological space, G a topological group and U= (Uᵢ)_i∈ I an open covering of B. Two cocycles (gᵢⱼ) and (g'ᵢⱼ) of Z¹_cont(B,U,G) are said to be cohomologous if there exi… 01RO
Proposition 11 Let B be a topological space, G a topological group and U= (Uᵢ)_i∈ I an open covering of B. 01RP
Theorem 4 The mapping r: H¹_cont(B,G)→ P(B,G) is bijective. 01RQ

§ 6. Simply connected spaces

Definition 1 A pointed set is a set X endowed with one of its elements, called the base point. The set X endowed with the point x is sometimes denoted by (X, x). Let (X, x) and (Y, y) be pointe… 01RR
Definition 2 A pointed covering (E, x) of (B, b) is said to be a universal covering if, for every pointed covering (E', x') of (B, b), there exists a unique morphism of coverings of B, f: E→ E'… 01RS
Proposition 1 Let B be a connected and locally connected topological space and let b be a point of B. Let (E, x) be a universal covering of (B, b). Then E is a Galois covering of B and every cov… 01RT
Corollary Let B be a locally connected topological space. Let b be a point of B and let (E, x) be a universal covering of (B, b). For a subspace A of B, the following two properties are equi… 01RU
Lemma 1 Let E be the n-dimensional numerical space and let A be a convex compact subset of E of which 0 is an interior point. For every x∈ E, let us denote by p_A(x) the greatest lower bou… 01RV
Proposition 2 Let E be the numerical space of dimension n and let A be a convex and compact subset of E of which 0 is an interior point. There exists a unique bijection u of E onto itself satisf… 01RW
Example The set [0,1]ⁿ is a convex, compact subset with nonempty interior of Rⁿ. It follows from proposition 2 of I, p. 123 that it is homeomorphic to the closed euclidean ball. More preci… 01RX
Definition 3 A topological space is said to be simply connected if all its coverings are trivializable. 01RY
Remark 1 Let B be a simply connected topological space and let (E, p) be a covering of B. If the space E is connected and nonempty, the mapping p is a homeomorphism. Let, for example, G be… 01RZ
Remark 2 Let B be a topological space such that every point has a simply connected neighbourhood; then every covering of a covering of B is a covering of B. Consider in fact a covering (E… 01S0
Proposition 3 Let B be a topological space. Let (E, p) be a covering of B and let y be a point of E. Let X be a simply connected topological space, let f: X→ B be a continuous mapping and let x… 01S1
Example 1 Let X be a simply connected topological space and let f be a continuous function from X into C^*. Recall (I, p. 101, example 6) that the mapping z↦ eᶻ makes C a covering of C^*. By… 01S2
Corollary Let B be a topological space and let b be a point of B. Let E be a simply connected covering of B. For every point x of E_b, the pointed space (E, x) is a universal covering of (B… 01S3
Proposition 4 The product of two simply connected spaces, one of which is locally connected, is a simply connected space. 01S4
Proposition 5 A connected and locally connected topological space such that the intersection of any two connected open subsets is connected is a simply connected space. 01S5
Corollary Every interval of the real line R is simply connected. 01S6
Example 2 The numerical space of n dimensions Rⁿ (TG, VI, p. 1) is simply connected. It is in fact a product of simply connected and locally connected spaces (I, p. 126, prop. 4 and cor. of… 01S7
Proposition 6 Let X be a topological space. Let U₁ and U₂ be open ( resp. closed) subspaces of X such that X = U₁∪ U₂. Suppose that U₁ and U₂ are simply connected and that their intersection U₁∩… 01S8
Example 3 For ngeqslant 2, the sphere Sₙ (TG, VI, p. 10) is simply connected. Indeed, the sphere Sₙ is the union of two closed hemispheres homeomorphic to Bₙ whose intersection is homeomorph… 01S9
Example 4 Let E be a finite-dimensional vector space of dimension n over R and let F be an affine subspace of E of codimension pgeqslant 3. The set Rᵖ-0 is homeomorphic to R× Sₚ₋₁ (TG, VI, p… 01SA
Proposition 7 Let X be a topological space. Let U₁ and U₂ be connected open ( resp. closed) subspaces of X such that X = U₁∪ U₂. If the space X is simply connected, then U₁∩ U₂ is connected. 01SB
Corollary Let X be a simply connected topological space and let A be a connected subset of X. If the complement of A is connected, its boundary is also connected. 01SC
Proposition 8 Let B be a topological space. Let T be a simply connected and locally connected space. Let E be a covering of B× T, with projection p, and let t be a point of T. Let Eₜ denote the… 01SD
Corollary 1 Under the hypotheses of prop. 8, if t and t' are two points of T, the coverings Eₜ and E_t' are B-isomorphic. 01SE
Corollary 2 Under the hypotheses of prop. 8, let (E, p) and (E', p') be coverings of B× T and let k: Eₜ→ E'ₜ be a B-morphism. There exists a unique (B× T)-morphism widetildek: E→ E' which exte… 01SF
Corollary 3 Let B and B' be topological spaces, let T be a simply connected and locally connected topological space. Let f: B'× T→ B be a continuous mapping and let E be a covering of B. Given… 01SG
Remark Let us retain the hypotheses and notations of proposition 8. Let G be a discrete topological group. Suppose that E is a principal covering of group G. If the coverings Eₜ of B and… 01SH
Proposition 9 Let B be a topological space and let (E, p) be a covering of B. Let T be a simply connected, locally connected and locally compact topological space. Let us denote by widetildep:C_… 01SI
Lemma Let X, Y, Z be topological spaces and let g: Y→ Z be a continuous mapping. 01SJ
Lemma 2 For every point x∈ X, the mapping μₓ: Eₓ× M→ X given by (g, u)↦ g· u is universally strict, and its image Eₓ· M is a neighbourhood of x in X. In particular, S· M is a neighbourhood… 01SK
Lemma 3 The group G is generated by S. 01SL
Lemma 4 The relation “(g₁, u₁) is congruent to (g₂, u₂)” is an equivalence relation in Z, compatible with the operation of F. 01SM
Lemma 5 If M is connected, the space Y is connected. 01SN
Lemma 6 For every x∈ X, the group N acts faithfully and transitively on the fibre overset-1p(x). 01SO
Lemma 7 Endowed with the action of N, the X-space (Y, p) is a principal left covering. It is trivializable over M. 01SP
Proposition 10 Let X be a nonempty connected topological space, let G be a discrete group operating continuously on the left in X, and let M be a subset of X such that G· M = X. One of the two fo… 01SQ
Proposition 11 Let X be a simply connected topological space and G a discrete group operating continuously on the right in X. If the subgroup of G generated by the union of the stabilizers of the… 01SR

II. Groupoids

§ 1. Quivers

Definition 1 A quiver is a quadruple (S,F, o, t), where S and F are sets and o,t are mappings of F into S. 01SS
Definition 2 Let C be a quiver. A path in C is a sequence c= (a₀, f₁, a₁, . . . , aₙ₋₁, fₙ, aₙ), where n is an integer geqslant 0,a₀, . . . , aₙ are vertices of C, and where, for 1leqslant ileq… 01ST

§ 2. Graphs

Definition 1 A graph[^1] is a quiver (S,F, o, t) endowed with an involution of F, denoted by f↦ overlinef, without fixed point and such that t(overlinef) =o(f) for every f∈ F. 01SU
Definition 2 A forest is a graph in which every loop without backtracking is a constant loop. A tree is a connected forest. 01SV
Proposition 1 Let G be a graph. Every forest of G is contained in a maximal forest of G; in particular, there exists a maximal forest of G. 01SW
Corollary A maximal forest of a connected graph is a maximal tree in it. 01SX
Remark 1 In LIE, IV, p. 33, appendix, the notion of a combinatorial graph was defined as a pair (A,S), where S is a set and A a subset of 𝔓(S) consisting of sets with two elements; the elem… 01SY

§ 3. Groupoids

Definition 1 Let C be a quiver. Let J denote the set of pairs (f, g) of arrows of C such that t(f) =o(g). A law of composition in C is a mapping m: J→ Fl(C) such that, for every pair (f, g)∈ J… 01SZ
Definition 2 A category is a quiver endowed with an associative law of composition for which there exists at each vertex an identity element. 01T0
Example 1 Let C be a quiver. Let Ch(C) denote the set of paths in C and o: Ch(C) → Som(C),t: Ch(C) → Som(C) the mappings which associate with a path its origin and its end. The quadruple Ω_C… 01T1
Example 2 Let Σ be a Species of structures in a theory T stronger than set theory and let σ (x, y, s, u) be a term of T satisfying conditions (MO_I), (MO_II) and (MO_III) of E, IV, p. 11. Le… 01T2
Definition 3 Let C and C' be categories. A functor from C into C' is a morphism of quivers φ : C→ C' such that φ (f g) = φ (f)φ (g) for every pair (f, g) of composable maps of C. 01T3
Definition 4 A groupoid is a category each arrow of which is invertible. 01T4
Proposition 1 Let G be a groupoid. The relation "there exists an arrow of G joining a to b" is an equivalence relation in the set of vertices of G whose equivalence classes are the orbits of G. 01T5
Definition 5 Let C be a category. A subcategory of C is a subquiver D of C satisfying the following conditions: 01T6
Definition 6 Let G be a groupoid. A subgroupoid of G is a subcategory H of G such that the inverse of every arrow of G belonging to H is an arrow of H. 01T7
Example 1 Let X be a set; let us denote by X× X the groupoid of pairs of X (II, p. 163, Example 2). Let R be an equivalence relation on X. The subquiver of the groupoid X× X whose set of ver… 01T8
Example 2 Let G and G' be groupoids, let φ and ψ be groupoid morphisms from G into G'. The equalizer of φ and ψ (cf. II, p. 153) is a subgroupoid of G. 01T9
Example 3 Let X be a set and Γ a group. Denote by X× Γ × X the graph whose set of vertices is X, whose set of arrows is X× Γ × X, the origin and end mappings being respectively pr₁ and pr₃… 01TA
Example 4 Let G be a groupoid, let X be a set and let φ : X→ Som(G) be a mapping. Define a graph G' in the following way. The set Som(G') is the set X; for every pair (x, y) of elements of X… 01TB
Example 5 Let φ : G→ G' be a morphism of groupoids. Let H be the subquiver of G with set of vertices Som(G) and whose set of arrows consists of the f∈ Fl(G) such that φ (f) is an identity el… 01TC
Example 6 Let φ : G→ G' be a morphism of groupoids. The quiver φ (G) whose set of vertices is φ(Som(G)) and whose set of arrows is φ(Fl(G)) is not in general a subgroupoid of G'. This is nev… 01TD
Example 1 Let G be a group, and let G be the groupoid associated with G (II, p. 163, Example 1). If X is a set, an operation (on the right) of the groupoid G on X is nothing other than an op… 01TE
Example 2 Let G = (S,F, o, t) be a groupoid. There exists a unique operation of G on the set S, relative to Id_S. It is given by φ (f)(a) =b if f∈ G_a,b. The orbits for this operation are th… 01TF
Definition 7 Let G be a groupoid. One says that a subgroupoid H of G is normal if Som(H) = Som(G) and if for every pair (a, b) of vertices of H and every arrow f∈ G_a,b, one has Int(f)(H_b) = H… 01TG
Lemma Given composable arrows u and v of G/H, one can choose representatives f and g of u and v in Fl(G) which are composable. The class modulo R of the product f g does not depend on th… 01TH
Definition 8 One says that G/H is the quotient groupoid of G by H and that p: G→ G/H is the canonical morphism. 01TI
Remark 1 The mapping Som(p) defines, by passing to quotients, a bijection of the set of orbits of G onto the set of orbits of G/H. In particular, G is transitive if and only if G/H is. 01TJ
Remark 2 Let a and b be vertices of G. The mapping of G_a,b into (G/H)_p(a),p(b) deduced from p is surjective. Let in fact u be an arrow of G/H joining p(a) to p(b); it is the class modulo… 01TK
Proposition 2 Let a be a vertex of G. The group homomorphism pₐ: Gₐ→ (G/H)ₚ₍ₐ₎ deduced from p by passing to the isotropy groups is surjective; its kernel is Hₐ. 01TL
Proposition 3 Let G be a groupoid, let H be a distinguished subgroupoid of G and let p: G→ G/H be the canonical morphism. Let φ : G→ G' be a groupoid morphism such that H⊂ Ker(φ ). There exists… 01TM
Proposition 4 Endowed with this law of composition, varpi_G is a groupoid. 01TN
Proposition 5 Let G be a graph; let us denote by j the canonical quiver morphism of G into varpi_G. Let φ be a quiver morphism of G into a groupoid G' such that φ (overlinef) =φ (f)⁻¹ for every… 01TO
Corollary 1 In a graph, every class of paths contains a unique path without retracing. 01TP
Corollary 2 The canonical morphism from G into varpi_G is injective. 01TQ
Corollary 3 Let G' be a subgraph of G. The morphism from varpi_G' into varpi_G deduced from the injection of G' into G is injective. 01TR
Corollary 4 A nonempty graph G is a tree if and only if the groupoid varpi_G is simply transitive. 01TS
Definition 9 Let C be a quiver. The groupoid varpi_widetildeC of classes of paths of the graph widetildeC associated with C is called the free groupoid constructed on C, and is denoted by Grp(C… 01TT
Proposition 6 Let C be a quiver, let G be a groupoid, and let φ be a quiver morphism from C into G. There exists a unique groupoid morphism φ ' from Grp(C) into G such that φ '∘ θ_C=φ. 01TU
Example Let C be a quiver having a unique vertex s and let A be the set of its arrows. The groupoid Grp(C) is then the groupoid associated with the free group F(A) constructed on A. In fac… 01TV
Proposition 7 For every groupoid morphism φ : G→ G' which maps every arrow belonging to F onto an identity element of G', there exists a unique groupoid morphism φ ': G/F→ G' such that φ '∘ p=φ. 01TW
Proposition 8 Let δ be the canonical morphism of the free groupoid Grp(Γ) into G extending the canonical injective morphism of Γ into G. Let a be a vertex of G and let A be the orbit of a in G… 01TX
Corollary 1 Suppose further that, for every arrow f of F, we have o(f) =t(f). Then the origin and the end of every path in the quiver Γ coincide. Let a be a vertex of G and let A be its orbit… 01TY
Corollary 2 If the graph associated with the quiver Γ is a forest, the homomorphism pₐ of Gₐ into (G/F)ₚ₍ₐ₎ is bijective, for every vertex a of G. 01TZ
Proposition 9 Let a be a vertex of G. There exists a unique homomorphism λ of the free group F(S-Fl(A)) into π₁(G, a) such that 01U0
Remark 1 The homomorphism μₐ is the inverse isomorphism of λ. 01U1
Remark 2 There exists a unique homomorphism λ : F(Fl(G)) → π₁(G, a) defined by relations (1) for every f∈ Fl(G). It follows from proposition 8 that the homomorphism λ is surjective and that… 01U2

§ 4. Homotopies

Definition 1 Let G be a groupoid, H a quiver, φ and φ ' morphisms of quivers from H into G. A homotopy connecting φ to φ ' is a mapping h of the set of vertices of H into the set of arrows of G… 01U3
Remark 1 Let G, G' be groupoids, H, H' quivers, let u: H'→ H be a morphism of quivers and v: G→ G' a morphism of groupoids. If morphisms of quivers φ ,φ ' of H into G are homotopic, the mor… 01U4
Remark 2 Let G be a groupoid, let H be a quiver, and let φ , ψ be morphisms of quivers of H into G. Let us denote by j the canonical morphism of H into Grp(H), and let overlineφ and overlin… 01U5
Definition 2 Let G, G' be groupoids and let φ be a morphism from G into G'. An inverse up to homotopy of φ is a groupoid morphism ψ from G' into G such that the morphisms ψ ∘ φ and φ ∘ ψ are re… 01U6
Proposition 1 Let G, G' be groupoids, let φ be a morphism from G into G', and let A be a subset of the set of vertices of G which meets every orbit of G. In order that φ be a homotopism, it is n… 01U7
Corollary 1 Let G be a groupoid, let A be a set and let f: A→ Som(G) be a mapping. If the image of f meets each orbit of G, the canonical groupoid morphism φ of f^*G into G is a homotopism. 01U8
Corollary 2 Let G be a groupoid, let X be an oriented forest of G, let G' be the groupoid deduced from G by contraction of the arrows of X. The canonical morphism of G into G' is a homotopism. 01U9
Proposition 2 The groupoid Coh(φ , ψ ) is generated by the subquiver whose set of vertices is Som(G) and whose set of arrows is the union of the images of the mappings Fl(α ) and h. 01UA
Proposition 3 The quiver morphism α is a groupoid morphism from G into Coh(φ , ψ ), and the mapping h is a homotopy joining α ∘ φ to α ∘ ψ. 01UB
Definition 3 The groupoid Coh(φ , ψ ) is called the cohomotopizer of the pair (φ , ψ ). One says that α is the canonical morphism from G into Coh(φ , ψ ) and that h is the canonical homotopy jo… 01UC
Proposition 4 The mapping Orb(α ): Orb(G) → Orb(Coh(φ , ψ )) is surjective; its fibres are the connected components of the framework of the pair (φ , ψ ). 01UD
Proposition 5 Let G' be a groupoid, let η ,η ' be groupoid morphisms of Coh(φ , ψ ) into G', and let k be a mapping of Som(G) into Fl(G'). In order that k be a homotopy joining η to η ', it is n… 01UE
Theorem 1 Make the following hypotheses: 01UF
Definition 4 An underlying equipment of the pair (φ , ψ ) is said to be given by the data: 01UG
Proposition 6 There exists a unique group homomorphism 01UH

§ 5. Coequalizer

Proposition 1 The group morphism β_a₀: G_a₀→ G'_β(a₀) is surjective, and its kernel is the smallest subgroup of G_a₀ containing the conjugacy classes c(z) for z∈ Ω (widetildeΓ). 01UI
Definition 1 A subset Z of Ω (widetildeΓ) is said to be distinguished (relative to the pair (φ , ψ )) if it satisfies the following properties: 01UJ
Proposition 2 If N is a distinguished subgroup of Gₐ, the set of elements z∈ Ω (widetildeΓ) such that c(z) is contained in N is a distinguished subset of Ω (widetildeΓ). 01UK
Corollary Let Z be a subset of Ω (widetildeΓ); in order that the conjugacy classes c(z), for z∈ Z, generate the kernel of the canonical homomorphism β_a₀: G_a₀→ G'_β(a₀), it is enough that t… 01UL
Definition 2 The groupoid Coeg(φ , ψ ) is called the coequalizer of the pair (φ , ψ ); the groupoid morphism γ is called the canonical morphism from G into Coeg(φ , ψ ). 01UM
Proposition 3 The pair (Coeg(φ , ψ ), γ ) has the following universal property: 01UN
Corollary The groupoid Coeg(φ , ψ ) is generated by the image of the morphism γ. 01UO
Remark 1 The set of vertices of Coeg(φ , ψ ) is the set of connected components of the quiver 01UP
Remark 2 The mapping Orb(γ ), deduced from γ by passing to orbits, defines by passage to the quotient a bijection of the set of connected components of the framework of the pair (φ , ψ ) on… 01UQ
Remark 3 The mapping Fl(γ ) of Fl(G) into Fl(Coeg(φ , ψ )) is not in general surjective. 01UR
Proposition 4 Let μ: H×_GH→ H be a quiver morphism such that φ ∘ μ=φ ∘ pr₁ and ψ ∘ μ=ψ ∘ pr₂. Suppose that, for every pair (a, b) of vertices of H such that φ (a) =φ (b), there exists a vertex c… 01US
Lemma 1 a) For every vertex a of Ker(φ , ψ ), ((a,1)) belongs to Z'₁. 01UT
Corollary a) With the notation of proposition 4, suppose moreover that the mapping (Som(φ ), Som(ψ )) of Som(H) into Som(G) × Som(G) is injective and that its image is the graph of an equiva… 01UU
Example Let X and R be topological spaces, let o and t be continuous mappings of R into X, let C be the set of pairs (f, g)∈ R² such that t(f) =o(g) in R, and let m be a continuous mapping… 01UV
Remark More generally, the hypotheses of the proposition are satisfied when (G,H, φ , ψ , μ) is a "groupoid of groupoids." This means that G and H are groupoids, φ and ψ are groupoid morp… 01UW
Definition 3 Let G and H be groupoids; let φ and ψ be morphisms of groupoids from H into G. Suppose that the groupoid Coeg(φ , ψ ) is transitive. Let (a, b, c₁, c₂,T, i₀) be a basic equipment o… 01UX
Proposition 5 Let G and H be groupoids; let φ and ψ be morphisms of groupoids from H into G. Suppose that the groupoid Coeg(φ , ψ ) is transitive. Let γ : G→ Coeg(φ , ψ ) denote the canonical mo… 01UY
Proposition 6 The unique group homomorphism 01UZ
Corollary 1 Suppose that the group K is generated by the stabilisers of the vertices of G. Then the group homomorphism γₒ: Gₒ→ (G/K)_γ(o) is surjective. Moreover, if the groupoid G is simply t… 01V0
Remark 1 One can give another description, sometimes more convenient, of the group (G/K)_γ(o). For this, let M = K× Gₒ and define a law of composition in M by the formula 01V1
Corollary 2 Suppose that the group K operates freely on Som(G). There then exists a unique group homomorphism π : (G/K)_γ(o)→ K whose kernel contains the image of γₒ and such that π (λ ([k]))… 01V2
Corollary 3 Suppose that the groupoid G is simply transitive. Let K₀ be the subgroup of K generated by the stabilizers of the vertices of G. The mapping from K into (G/K)_γ(o) which to k∈ K as… 01V3

III. Homotopy and Poincaré groupoid

§ 1. Homotopies, homeotopies

Definition 1 Let X and Y be topological spaces and let f and g be continuous mappings of X into Y. A homotopy relating f to g is by definition a continuous mapping σ : X× I→ Y such that, for ev… 01V4
Proposition 1 Let X and Y be topological spaces. The relation “ f is homotopic to g ” is an equivalence relation in the set C(X; Y) of continuous mappings of X into Y. 01V5
Proposition 2 Let X, Y and Z be topological spaces, f and f' continuous mappings of X into Y,g and g' continuous mappings of Y into Z. If f is homotopic to f' and if g is homotopic to g', then g… 01V6
Proposition 3 Let X be a topological space and let (Yⱼ)_j∈ J be a family of topological spaces. The mapping of [X;∏_j∈ JYⱼ] into the product set ∏_j∈ J[X; Yⱼ] defined by φ ↦ ([prⱼ]∘ φ )_j∈ J is… 01V7
Corollary Let (Xⱼ)_j∈ J and (Yⱼ)_j∈ J be families of topological spaces having the same index set. For each j∈ J, let fⱼ and gⱼ be continuous mappings of Xⱼ into Yⱼ. If, for each j∈ J, the m… 01V8
Definition 2 Let X and Y be topological spaces. A homotopy class φ ∈ [X; Y] is said to be invertible if there exists a homotopy class ψ ∈ [Y; X] such that ψ ∘ φ = [Id_X] and φ ∘ ψ = [Id_Y]. A c… 01V9
Definition 3 One says that topological spaces X and Y are homotopy equivalent if there exists a homotopy equivalence of X into Y. 01VA
Example 1 The empty topological space is homotopy equivalent only to itself. 01VB
Example 2 Let X be a nonempty topological space. In order that X be homotopy equivalent to a space reduced to one point, it is necessary and sufficient that there exist a constant mapping p… 01VC
Example 3 One says that a pointed topological space (X, x) is contractible, or that the topological space X is contractible at x, if there exists a homotopy pointed at x connecting Id_X to t… 01VD
Example 4 Let E be the numerical space of n dimensions (or, more generally, a topological vector space over R) and let X be a subset of E. One says that the set X is star-shaped at a point x… 01VE
Example 5 We shall prove later (TA, V) that the Euclidean spheres Sₙ,ngeqslant 1, are not spaces homotopic to a point. The unit sphere of a Hilbert space of countable type and infinite dimen… 01VF
Definition 4 Let X be a topological space and let A be a subset of X. 01VG
Lemma 1 Let X and Y be topological spaces and let f: X→ Y be a continuous mapping. The following conditions are equivalent: 01VH
Definition 5 Let X be a topological space and let A be a subset of X. A contraction of X onto A is a homotopy σ : X× I→ X fixed on A whose origin is the identity mapping of X and whose term is… 01VI
Remark In other words, a contraction of X onto A is a homotopy σ : X× I→ X having the following properties: 01VJ
Definition 6 With the preceding notations, one says that σ is a strong contraction if, moreover, one has r(σ (x, t)) =r(x) for every x∈ X and every t∈ I, 01VK
Example Let X be the complement of the origin in Bₙ. The mapping of X× I into X given by (x, t)↦ ((1-t) +t1/|x|)x is a strong contraction of X onto Sₙ₋₁. The retraction of X onto Sₙ₋₁ asso… 01VL
Lemma 2 Let X be a topological space, let U be an open subset of X and let σ be a strong contraction of X onto X-U. Then σ (U× I)⊂ overlineU. In particular, σ (U× 1) is contained in the bo… 01VM
Definition 7 The cylinder of the mapping f is called and is denoted by Cyl(f), the topological space quotient U/R. 01VN
Example 1 Let X',Y' be topological spaces and let f': X'→ Y' be a continuous mapping. Let u: X→ X' and v: Y→ Y' be continuous mappings such that f'∘ u=v∘ f. There exists a unique continuous… 01VO
Proposition 5 The mapping α_f induces a homeomorphism of X× [0,1[ onto an open subset of Cyl(f). The mapping β_f defines a homeomorphism of Y onto the complement of this open subset. The mapping… 01VP
Proposition 6 The mapping σ_f: Cyl(f)× I→ Cyl(f) is a strong contraction of Cyl(f) onto its base. Its term is the canonical retraction of Cyl(f) onto its base. 01VQ
Remark 1 Let A be a subset of X× I and let A₁ be the set of points x∈ X such that (x,1)∈ A. We have α⁻¹_f(α_f(A)) = A∪ overset-1f(f(A₁))× 1 and β⁻¹_f(α_f(A)) =f(A₁). It follows that the map… 01VR
Remark 2 Suppose that the mapping f is proper. Let P be a point of Cyl(f). If P = α_f(x, t), with 0 leqslant t < 1 and x∈ X, we have α⁻¹_f(P) = (x, t). In the contrary case, there exists y∈… 01VS
Remark 3 If the spaces X and Y are separated, the same is true of the mapping cylinder of the mapping f. Indeed, let z and z' be two distinct points of Cyl(f) and let us prove that they pos… 01VT
Definition 8 Let X be a topological space and let A be a subset of X. We say that the pair (X,A) possesses the homotopy extension property if, for every topological space Y, every continuous ma… 01VU
Remark 1 Let X be a topological space and let A be a subset of X such that the pair (X,A) possesses the homotopy extension property. Let Y be a topological space, let f: X→ Y be a continuou… 01VV
Proposition 7 With the above notations, the following assertions are equivalent: 01VW
Remark 2 Let X be a separated topological space and let A be a subset of X such that the pair (X,A) has the homotopy extension property. Let us denote by i: A→ X the canonical injection. Le… 01VX
Lemma 3 Let X and Y be topological spaces, let p: X→ Y be a continuous proper and open mapping, and let f: X→ R be a continuous mapping. The mapping g: Y→ overlineR given by y↦ sup_x∈overs… 01VY
Theorem 1 Let X be a topological space and let A be a closed subspace of X; denote by i: A→ X the canonical injection. The following assertions are equivalent: 01VZ
Lemma 4 Let X and Y be topological spaces, let φ : X→ R₊ be a continuous mapping, and put A =overset-1φ(0). Let σ : X× I→ Y be a homotopy which is fixed on A. The mapping σ ': X× I→ Y whic… 01W0
Corollary 1 Let X be a normal topological space and let A be a closed subspace of X. Suppose that there exist a neighbourhood V of A in X and a contraction of V onto A, together with a continu… 01W1
Example 1 Take for space X the ball Bₙ and for subspace A the sphere Sₙ₋₁. If V = X-0, there exists a strong contraction of V onto Sₙ₋₁(III, p. 237, example). Hence the pair (Bₙ,Sₙ₋₁) posses… 01W2
Corollary 2 Let X and Y be topological spaces, let A be a closed subspace of X, let B be a closed subspace of Y. If the pairs (X,A) and (Y,B) possess the homotopy extension property, the same… 01W3
Example 2 Here are other cases in which the pair (X,A) possesses the homotopy extension property. 01W4
Definition 9 The quotient topological space Y/R is called the space obtained by attaching the space B to the space X along the mapping f. It is denoted by X∪_fB. 01W5
Remark 1 The mapping α_f is injective. For every subset U 01W6
Remark 2 Let V be a subset of B. One has α⁻¹_f(β_f(V)) =f(V∩ A) and 01W7
Remark 3 Suppose that A is closed and that the mapping f is proper. We have just seen that the mapping β_f is closed. For every x∈ X, 01W8
Example 1 Let X and Y be topological spaces and let f: X→ Y be a continuous mapping. The cylinder Cyl(f) is none other than the space obtained by attaching the space Y to the space X× I alon… 01W9
Example 2 Let X be a topological space, let B be a topological space and let A be a subspace of B, and let f: A→ X be a continuous mapping inducing a homeomorphism of A onto its image. 01WA
Proposition 8 Let Z be a topological space. 01WB
Corollary 1 Let B' be a topological space, let A' be a subspace of B', and let f': A'→ X be a continuous mapping. Let v: B→ B' be a continuous mapping such that v(A)⊂ A' and f = f'∘ (v|A). Let… 01WC
Example 3 Let i be the canonical injection of A into B and let us take for the space B' the cylinder of the mapping i; let αᵢ: A× I→ Cyl(i) and βᵢ: B→ Cyl(i) be the canonical mappings and rᵢ… 01WD
Corollary 2 Let X' be a topological space, let u: X→ X' be a continuous mapping, and put f'=u∘ f. Let w: X∪_fB→ X'∪_f'B be the unique continuous mapping such that w∘ α_f=α_f'∘ u and w∘ β_f=β_f… 01WE
Proposition 9 Let X and B be topological spaces, let A be a subspace of B, and let f: A→ X be a continuous mapping. 01WF
Definition 10 The quotient space of X by R is denoted by X/A and is called the space obtained from X by contraction of the subset A. 01WG
Proposition 10 Let X be a topological space and let A be a subset of x. Suppose that there exists a homotopy σ : X× I→ X with origin Id_X, constant on A× 1 and such that σ (A× I)⊂ A. Then the can… 01WH
Remark Let X be a topological space and let A be a subset of X. 01WI
Definition 11 The cone of the mapping f is the topological space deduced from Cyl(f) by contracting f₀(X); it is denoted by Côn(f). 01WJ
Example 1 Let X be a topological space. One denotes by C(X), and calls the cone of the space X, the cone of the mapping Id_X; it is the space obtained from X× I by contracting X× 0. Let π be… 01WK
Remark 1 Let X be a topological space and let A be a subspace of X; let i denote the canonical injection of A into X. The canonical retraction rᵢ of the cylinder Cyl(i) onto its base maps t… 01WL
Remark 2 Let X and Y be topological spaces and let f: X→ Y be a continuous mapping. The canonical mapping α '_f: X× I→ Cone(f) is constant on X× 0 and therefore defines a continuous mapping… 01WM

§ 2. Homotopy and paths

Definition 1 Let X be a topological space. A path in X is called any continuous mapping c of I into X. The point c(0) is called the origin, the point c(1) the term of the path c. A loop in X is… 01WN
Definition 2 Let X be a topological space. One says that paths c and d in X are juxtaposed if one has c(1) =d(0). One then calls juxtaposed path of c and d the path c*d defined by the formula: 01WO
Remark 1 Let X be a topological space and let P be a topological space reduced to a point. Identify P× I with I by the projection pr₂. The set C(P× I; X) of homotopies between (necessarily… 01WP
Remark 2 Let X and Y be topological spaces. The canonical mapping 01WQ
Proposition 1 Let X and Z be topological spaces. For a mapping φ of Z into the space of paths C_c(I; X) to be continuous, it is necessary and sufficient that the mapping (z, t)↦ φ (z)(t) be a co… 01WR
Proposition 2 Let X be a topological space. The map which associates with a path c the opposite path overlinec is a homeomorphism of Λ(X) onto itself. The juxtaposition of paths (c, d)↦ c*d is a… 01WS
Corollary Let X be a topological space and let x,y,z be points of X. The mappings c↦ overlinec of Λ_x,y(X) into Λ_y,x(X) and (c, d)↦ c*d of Λ_x,y(X)× Λ_y,z(X) into Λ_x,z(X) are continuous. 01WT
Definition 3 A topological space X is said to be arcwise connected if for every pair (x, y) of points of X, there exists a path of origin x and of term y. 01WU
Example 1 Every interval of R, every convex subset of a numerical space or, more generally, of a topological vector space over R is arcwise connected. 01WV
Proposition 3 The image by a continuous mapping of an arcwise connected space is arcwise connected. 01WW
Proposition 4 A path-connected topological space is connected. 01WX
Proposition 5 In a topological space, the relation “there exists a path with origin x and term y” is an equivalence relation. 01WY
Example 2 The union of a family of path-connected sets, whose intersection is nonempty, is path-connected (cf. TG, I, p. 81, prop. 2). 01WZ
Example 3 A subset of a numerical space (or, more generally, a topological vector space over R) which is star-shaped at one of its points is path-connected. 01X0
Proposition 6 Let (Yⱼ)_j∈ J be a family of topological spaces. The mapping 01X1
Definition 4 A topological space is said to be locally path-connected if each of its points possesses a fundamental system of path-connected neighbourhoods. 01X2
Proposition 7 A locally path-connected topological space is locally connected. Its connected components coincide with its path-connected components. In particular, if a locally path-connected to… 01X3
Corollary 1 Every connected open subset of a locally path-connected topological space is path-connected. 01X4
Corollary 2 Let B be a locally path-connected topological space and let E be a B-space étalé. The space E is locally path-connected. If it is connected, it is path-connected. 01X5
Proposition 8 Every quotient space of a locally path-connected space is locally path-connected. 01X6
Proposition 9 The product of a family of locally path-connected spaces, connected with the exception of a finite number of them, is locally path-connected. 01X7
Corollary Every open subset of a numerical space is locally path-connected. 01X8
Example Let G be the subgroup of GL(n,R) formed by the square matrices of order n whose determinant is strictly positive. The group G is connected and locally path-connected. 01X9
Proposition 10 Let X be a topological space. The mapping (o, e) of Λ (X) =C_c(I; X) into X× X, which associates with a path c in X the pair (c(0), c(1)), is continuous. If the space X is locally… 01XA
Lemma Let c:I→ X be a path and let W be a neighbourhood of c in the space C_c(I; X). There exists a real number ε ∈ ]0,1/2], a neighbourhood V₀ of c(0) and a neighbourhood V₁ of c(1) in… 01XB
Corollary Soit X a topological space locally connected by arcs and let x be a point of X. The mapping c↦ c(1) of Λₓ(X) into X is open. 01XC
Proposition 11 A connected and locally connected topological space, whose topology can be defined by a distance for which it is complete, is connected by arcs. 01XD
Lemma 1 Let X be a connected and locally connected metric space, let a and b be points of X and let ε be a real number >0. There exists in X a train of width leqslant ε joining a to b. 01XE
Corollary 1 A locally connected topological space, whose topology can be defined by a distance for which it is complete, is locally arcwise connected. 01XF
Lemma 2 Let X be a complete metric space and let U be an open subset of X. There exists a distance on U compatible with its topology for which U is complete. 01XG
Corollary 2 A locally compact, locally connected and metrizable topological space is locally arcwise connected. If it is connected, it is arcwise connected. 01XH
Definition 5 Let X and Y be topological spaces. A mapping f: X→ Y is said to be arcwise continuous if, for every path c in X, the mapping f∘ c:I→ Y is continuous. 01XI
Remark Suppose that the space X is arcwise connected. Let x be a point of X. For f to be arcwise continuous, it suffices that, for every path c with origin x, the mapping f∘ c be continuo… 01XJ
Proposition 12 Let X and Y be topological spaces and let f: X→ Y be a mapping. Suppose that the space X is locally arcwise connected and that every point of X possesses a countable fundamental sy… 01XK
Lemma Let X be a topological space connected and locally connected by arcs and let x be a point of X. Suppose that the point x has a countable fundamental system of neighbourhoods. Then… 01XL
Proposition 13 Let p: E→ B be an étale and separated mapping and let s: B→ E be a section of p. If the space B is locally path-connected and if the section s is path-continuous, it is continuous. 01XM
Corollary Let B be a topological space, let (E, p) and (E', p') be two B-spaces. Suppose that the mapping p is étale and separated and that the space E' is locally path-connected. Then every… 01XN
Proposition 14 Every compact, metrizable, nonempty, totally disconnected topological space without isolated point is homeomorphic to 0,1^N. 01XO
Lemma 3 Let ε be a real number >0. There exists an integer mgeqslant 1 such that, for every integer ngeqslant m, X admits a partition formed of n nonempty open and closed sets of diameter… 01XP
Example Let K be the triadic Cantor set (TG, IV, p. 9, example). For every ngeqslant 0, put Jₙ=0,1 and define a mapping Kₙ from the set Cₙ= J₀× ⋯ × Jₙ into the set of closed intervals of [… 01XQ
Corollary Let X be a metrizable, compact and nonempty topological space. There exists a continuous and surjective mapping from 0,1^N into X. 01XR
Proposition 15 The image of a path in a separated topological space is a compact, metrizable, connected and locally connected by arcs topological space. 01XS
Theorem 1 (Hahn and Mazurkiewicz) Every metrizable, compact, nonempty, connected and locally connected topological space is homeomorphic to a quotient space of the segment [0,1]. 01XT
Lemma 4 Let K be a compact topological space and let R be a set of open subsets of K covering K. There exists an entourage V of the uniform structure of K (TG, II, p. 27, th. 1) such that… 01XU
Lemma 5 Let X and Y be uniform spaces and f a mapping of X into Y. Let F be a set of closed subsets of X covering X and possessing the following properties: 01XV
Lemma 6 Let D be a countable totally ordered set, not reduced to one element, possessing a smallest and a greatest element. Suppose that D has no gap (E, III, p. 73, exerc. 19), that is to… 01XW
Proposition 16 Let E be a totally ordered set not reduced to one element. Suppose that every subset of E has a supremum and that there exists a countable subset of E which meets every open interv… 01XX
Remark Let E be a totally ordered set. The set of open intervals of E (bounded or not) is stable under finite intersection. It is a basis of a topology T₀(E) on E (TG, I, p. 91, exerc. 5)… 01XY
Corollary Let R be an equivalence relation in I. The following conditions are equivalent: 01XZ
Proposition 17 Let X be a connected and compact topological space. Let a be a point of X, let U be a nonempty open and closed subset of X -a. 01Y0
Corollary Let X be a compact connected topological space, let N be the set of points of X whose complement is connected. The set X is the unique connected and compact subset of X which conta… 01Y1
Lemma 7 Soit T un espace topologique localement connexe. Soit U un ensemble filtrant croissant de parties ouvertes de T, de réunion T. Pour t∈ T et U∈ U, notons Uₜ la composante connexe de… 01Y2
Theorem 2 Let X be a compact connected topological space possessing a countable everywhere dense subset. Let a,b be distinct points of X. The following conditions are equivalent: 01Y3
Proposition 18 Let X be a separated topological space. Let a and b be distinct points of X which belong to the same arcwise connected component of X. There exists an injective path joining a to b… 01Y4
Lemma 8 The set U, ordered by inclusion, is inductive. 01Y5
Lemma 9 There exists a continuous, increasing and surjective mapping u:I→ I and an injective path c:I→ X joining a to b such that g=c∘ u. 01Y6
Theorem 3 Let I be an interval of R and let X be a nonempty and separated topological space. Let p: X→ I be a continuous, open and proper mapping whose fibres are totally disconnected (TG, I… 01Y7
Lemma 10 Let a,b be points of I such that aleqslant b. 01Y8
Lemma 11 a) The set K' is not empty. 01Y9
Corollary Let X and B be topological spaces and let p: X→ B be a continuous, open, proper and separated mapping whose fibres are totally disconnected. Let I be an interval of R, let f: I→ B… 01YA
Theorem 4 Let X be a topological space, let G be a discrete group operating properly in X and let p: X→ X/G be the canonical mapping. Let I be an interval of R, let f: I→ X/G be a continuous… 01YB
Example 1 Let X be a separated topological space and let G be a finite group, endowed with the discrete topology, which operates continuously in X. The operation is then proper (TG, III, p… 01YC
Example 2 Soit n an integer geqslant 0. Let Pₙ denote the set of monic polynomials P∈ C[X] of degree n, endowed with the topology for which the mapping (c₀, . . . , cₙ₋₁)↦ Xⁿ+cₙ₋₁Xⁿ⁻¹+⋯ +c₀… 01YD

§ 3. Poincaré groupoid

Definition 1 Let X be a topological space, let c₀ and c₁ be paths in X and σ :I× I→ X a homotopy joining c₀ to c₁. One says that σ is a strict homotopy if the mappings s↦ σ (0, s) and s↦ σ (1… 01YE
Example Let c be a path in X and let φ :I→ I be a continuous mapping such that φ (0) = 0 and φ (1) = 1. The paths c and c∘ φ are strictly homotopic. In fact, the mapping σ :I× I→ X defined… 01YF
Definition 2 One calls the space of loops of X, and denotes by Ω(X), the subspace of Λ(X) consisting of loops (III, p. 256, def. 1) in X. 01YG
Proposition 1 Let X be a topological space, let x,y,z,u be points of X, and let γ₁∈ varpi_x,y(X),γ₂∈ varpi_y,z(X),γ₃∈ varpi_z,u(X) be path classes. One has 01YH
Remark 1 Let X be a topological space. Let n be an integer geqslant 1 and let (c₁, . . . , cₙ) be a sequence of paths in X such that cᵢ and cᵢ₊₁ are juxtaposable for 1leqslant ileqslant n-1… 01YI
Definition 3 Let X be a topological space. The groupoid varpi (X) is called the Poincaré groupoid, or fundamental groupoid, of the space X. Let x be a point of X; the group π₁(X, x) of classes… 01YJ
Proposition 2 Let X be a topological space. 01YK
Remark 2 Let X be a topological space and let C be an arcwise connected component of X. If x and y are points of C, the topological space Λ_x,y(C) is identified with the topological space Λ… 01YL
Remark 3 Let X be a topological space and let x,y,z be points of X. The mapping (c, d)↦ c*d of Λ_x,y(X)× Λ_y,z(X) into Λ_x,z(X) is continuous (III, p. 258, corollary of Prop. 2). It should… 01YM
Remark 1 The homomorphism varpi_x,y(f) is continuous if one endows the sets varpi_x,y(X) and varpi_f(x),f(y)(Y) with the quotient topology of the topology of compact convergence on C_c(I; X… 01YN
Proposition 3 Let X and Y be topological spaces, f₀ and f₁ continuous mappings of X into Y, and let σ : X× I→ Y be a homotopy joining f₀ to f₁. Let x be a point of X; put y₀=f₀(x), y₁=f₁(x), and… 01YO
Lemma 1 Let Y be a topological space and let φ :I× I→ Y be a continuous mapping. For t∈ I, put c₀(t) =φ (t,0),c₁(t) =φ (t,1), d₀(t) =φ (0, t) and d₁(t) =φ (1, t). The paths c₀*d₁ and d₀*c₁… 01YP
Corollary 1 Let X and Y be topological spaces and let x be a point of X. Let f₀ and f₁ be continuous mappings of X into Y. If there exists a pointed homotopy at x joining f₀ to f₁, one has π₁(… 01YQ
Corollary 2 Let X and Y be topological spaces and let f: X→ Y be a homeomorphism. For every point x of X, the homomorphism 01YR
Example Let G be a topological group, let e be its identity element. For every point g of G, the left and right translations, x↦ gx and x↦ xg, are homeomorphisms of G onto itself (TG, III… 01YS
Corollary 3 Let X be a topological space homotopic to a point. For every point x of X, the group π₁(X, x) is reduced to the identity element. 01YT
Proposition 4 Let X be the product space of a family (Xⱼ)_j∈ J of topological spaces. The morphism of the groupoid varpi (X) into the product of the groupoids varpi (Xⱼ), for j∈ J, defined by th… 01YU
Corollary Let x= (xⱼ)_j∈ J be a point of X. The mapping 01YV
Remark 2 Let (Xⱼ)_j∈ J be a family of topological spaces. Let X denote the product topological space ∏_j∈ JXⱼ and let x= (xⱼ) be a point of X. 01YW
Remark 3 Let (Xⱼ)_j∈ J be a family of topological spaces. Let X denote the product topological space ∏_j∈ JXⱼ and let x= (xⱼ) be a point of X. 01YX
Proposition 5 Let X be a topological space and let (Aᵢ)_i∈ I be an increasing family of subsets of X, indexed by a filtered ordered set I, such that every quasi-compact subset of X is contained… 01YY
Corollary Let a be a point of X and let J be the set of i∈ I such that a∈ Aᵢ. The canonical homomorphism 01YZ
Remark 4 The proposition and its corollary apply in particular when (Aᵢ)_i∈ I is an increasing family of subsets of X indexed by a directed ordered set I such that the interiors of the Aᵢ c… 01Z0
Definition 4 Let X be a topological space and let c and c' be two loops in X. A free homotopy joining c to c' means a homotopy σ joining c to c' such that σ (0, s) =σ (1, s) for every s∈ I. One… 01Z1
Remark Let φ be the canonical mapping of R onto T=R/Z (TG, V, p. 2). The mapping f↦ f∘ φ |I is a homeomorphism of C_c(T; X) onto Ω(X), whence, by passing to arcwise connected components… 01Z2
Proposition 6 Let X be an arcwise connected topological space and let x be a point of X. 01Z3
Scholium Let X be an arcwise connected topological space and let x be a point of X. Proposition 6 makes it possible to define a canonical bijection of the set of free homotopy classes of lo… 01Z4

§ 4. Homotopy and coverings

Proposition 1 Let B be a topological space and let E be a covering of B. Let B' be a topological space and let f₀ and f₁ be continuous mappings of B' into B. If the mappings f₀ and f₁ are homoto… 01Z5
Corollary A topological space homotopic to a simply connected topological space is simply connected. 01Z6
Remark Let G be a discrete topological group. With the notation of the proposition, suppose that E is a principal covering with group G. Then the coverings f₀^*(E) and f₁^*(E) are isomorp… 01Z7
Proposition 2 (Lifting of homotopies) Let B be a topological space and let E be a covering of B. Let B' be a topological space and let f₀ and f₁ be continuous mappings of B' into B. Let σ : B'× I→ B be a homotopy joini… 01Z8
Proposition 3 Let B be a topological space and let p: E→ B be a covering. Let us denote by widetildep:C_c(I; E)→ C_c(I; B) the mapping c↦ p∘ c. Let us denote by o_E:C_c(I; E)→ E ( resp. o_B:C_c(… 01Z9
Corollary 1 The mapping widetildep:C_c(I; E)→ C_c(I; B) is a covering. 01ZA
Corollary 2 (Lifting of paths) Let p: E→ B be a covering, let x be a point of E and a=p(x). The mapping c↦ p∘ c is a homeomorphism of the space Λₓ(E) of paths in E with origin x onto the space Λₐ(B) of paths in… 01ZB
Proposition 4 Let p: E→ B be an étale and separated mapping which satisfies the path-lifting property. Let a and b be points of B and let x be a point of E such that p(x) =a. Let c₀ and c₁ be tw… 01ZC
Corollary 1 Let B be a topological space and let E be a covering of B; let us denote by p its projection. For every pair (x, y) of points of E, the mapping varpi_x,y(p):varpi_x,y(E)→ varpi_p(x… 01ZD
Corollary 2 Let B be a topological space and let E be a covering of B; let us denote by p its projection. Let x be a point of E, and put b=p(x). In order that the class in π₁(B, b) of a loop c… 01ZE
Corollary 3 Let B be a topological space and let (E₁, p₁) and (E₂, p₂) be coverings of B. Let us denote by (E, p) their fibre product over B, and let x= (x₁, x₂) be a point of E. The image of… 01ZF
Remark Let b and b' be two points of B,x a point of E_b,c∈ Λ_b,b'(B) a path in B and widetildec the path in E originating at x which lifts c. For every s∈ I, let us denote by cₛ the path… 01ZG
Theorem 1 Let B be a topological space, let (E, p) be a covering of B, let b be a point of B and let x be a point of the fibre E_b. 01ZH
Proposition 5 The mapping h_(E,x) of π₁(B, b) into G which, to γ ∈ π₁(B, b), associates the unique element g of G such that x· g=x· γ⁻¹, is a group homomorphism, whose kernel is the subgroup p_*… 01ZI
Example 1 Let F be a discrete set endowed with an operation of G, and let X = E×ᴳF be the associated covering of B. Let us denote by φ : E× F→ X the canonical B-morphism. It is a morphism of… 01ZJ
Example 2 Let H be a discrete topological group, let f: G→ H be a group morphism. Endow H with the left operation of G given by g· h=f(g)h, for g∈ G and h∈ H. Let E'= E×ᴳH be the associated… 01ZK

§ 5. Homotopy and coverings (case of locally arcwise connected spaces)

Proposition 1 Let B be a topological space and let (E, p) be a covering of B. Let Y be a topological space and let f: Y→ B be a continuous mapping. Let y∈ Y,x∈ E,b∈ B be points such that f(y) =p… 01ZL
Corollary 1 Let B be a connected topological space locally connected by paths, and let b be a point of B. In order that a covering (E, p) of B be trivializable, it is necessary and sufficient… 01ZM
Remark Let B be a topological space locally connected by paths, let b be a point of B, and let V be an open connected neighbourhood of b such that the image of the homomorphism from π₁(V… 01ZN
Corollary 2 Let B be a connected topological space locally connected by paths. If, for a point b of B, the group π₁(B, b) is reduced to the identity element, the space B is simply connected. 01ZO
Corollary 3 Let 01ZP
Proposition 2 Let E and E' be coverings of B. 01ZQ
Corollary 1 The coverings E and E' are isomorphic if and only if the π₁(B, b)-sets E_b and E'_b are isomorphic. 01ZR
Corollary 2 Let (E, p) and (E', p') be connected coverings of B. Let x be a point of E_b and x' a point of E'_b. In order that there exist a B-morphism g: E→ E' such that g(x) =x', it is neces… 01ZS
Corollary 3 Let (E, p) be a connected covering of B and let x be a point of the fibre E_b. Let N denote the normalizer of p_*(π₁(E, x)) in π₁(B, b). For every element γ of N, there exists a un… 01ZT
Corollary 4 Let (E, p) be a connected covering of B and let x be a point of the fibre E_b. In order that E be a Galois covering of B, it is necessary and sufficient that p_*(π₁(E, x)) be a dis… 01ZU
Lemma 1 Let B be a topological space locally connected by paths, let p: E→ B and p': E'→ B be étale and separated mappings satisfying the path-lifting property (III, p. 302). Let g: E→ E'… 01ZV
Remark Let B be a topological space locally connected by paths and suppose that every point b of B possesses a neighbourhood V such that the image of π₁(V, b) in π₁(B, b) reduces to the i… 01ZW
Proposition 3 Let B be a topological space locally path-connected and let E be a set endowed with an operation without local monodromy φ of the groupoid varpi (B), relative to a mapping p: E→ B… 01ZX
Corollary Let B be a connected topological space, locally connected by arcs, let (E, p) be a B-space whose projection p is etale, separated, and has the path-lifting property. If the canonic… 01ZY
Proposition 4 There exists a unique topology on π₁(B, a), compatible with its group structure, for which the admissible normal subgroups of π₁(B, a) form a fundamental system of neighbourhoods o… 01ZZ
Remark 1 If B₀ denotes the path-connected component of a in B, the canonical isomorphism π₁(B₀, a)→ π₁(B, a) is an isomorphism of topological groups when these groups are endowed with the a… 0200
Remark 2 Let B be a topological space. Let b and b' be points of B which belong to the same path-connected component and let γ be an element of varpi_b,b'(B). It follows from the definition… 0201
Remark 3 Let A and B be topological spaces, let a be a point of A, and let f: A→ B be a continuous mapping. Put b=f(a). If H is an admissible subgroup of π₁(B, b), its inverse image under t… 0202
Proposition 5 Let B be a topological space locally arcwise connected, and let a be a point of B. In order that a subgroup H of π₁(B, a) be admissible, it is necessary and sufficient that there e… 0203
Proposition 6 Let B be a topological space and let b be a point of B. Endow the group π₁(B, b) with the admissible topology. 0204
Proposition 7 Let B be a connected topological space, locally connected by arcs, and let b be a point of B. Endow the group π₁(B, b) with the admissible topology. 0205
Remark 4 The admissible topology on π₁(B, b) is the least fine topology for which the action of π₁(B, b) on E_b is continuous for every connected covering E of B (Prop. 6 of III, p. 318 and… 0206
Remark 5 Let B be a connected and locally arcwise connected topological space, let a be a point of B, and let H be a subgroup of π₁(B, a). By the preceding remark, if H is admissible, it is… 0207

IV. Contractible Spaces

§ 1. Simply connected spaces

Definition 1 A topological space X is said to be arcwise simply connected if it is arcwise connected and if every loop in X is strictly homotopic to a constant loop. 0208
Definition 2 A topological space B is said to be unloopable if it is locally arcwise connected and if every point b of B possesses a neighbourhood V such that the homomorphism of π₁(V, b) into… 0209
Remark 1 There exist topological spaces B, connected and locally arcwise connected, such that certain points a of B, but not all, possess a neighbourhood V such that the homomorphism of π₁(… 020A
Remark 2 Let B be an unloopable space; every operation of the groupoid varpi (B) on a B-space is without local monodromy (cf. III, p. 313, remark). In particular, in order that a mapping p… 020B
Proposition 1 The product space of a finite family of unloopable spaces is unloopable. 020C
Proposition 2 a) Every covering of an unloopable space is unloopable. 020D
Proposition 3 Let B be an unloopable topological space. Let (Y, q) be a covering of B and let (Z, p) be a covering of Y. The topological space Z, endowed with the mapping q∘ p, is then a coverin… 020E
Theorem 1 Let B be a connected topological space, locally connected by arcs, and let b be a point of B. The following properties are equivalent: 020F
Remark 1 Let B be a connected unwindable space and let b be a point of B. It follows from Theorem 1, (iv), that every operation of the group π₁(B, b) is admissible. Consequently, every π₁(B… 020G
Corollary 1 Let B be an unwindable topological space and let b be a point of B. Let E be a connected covering of B and let x be a point of the fibre E_b. The following properties are equivalen… 020H
Corollary 2 Let B be an unwindable topological space and let b be a point of B. Let (E, x) be a universal covering of the pointed space (B, b). For every covering E' of B and every point x' of… 020I
Proposition 4 Let B be an unwindable topological space. The relation of strict homotopy is an open equivalence relation in the topological space C_c(I; B). 020J
Definition 3 A topological space B is said to be locally contractible if every point b of B possesses a neighbourhood V such that the pointed space (V, b) is contractible (III, p. 234, example… 020K
Proposition 5 A locally contractible topological space is unlaceable. 020L
Proposition 6 The mapping p:x↦ e^2π ix of R into S₁ makes (R,0) a universal covering of (S₁,1). The group π₁(S₁,1) is isomorphic to Z; the class of the loop t↦ e^2π it is a generator of it. For… 020M
Corollary The mapping z↦ eᶻ of C into C^* makes (C,0) a universal covering of (C^*,1). The group π₁(C^*,1) is isomorphic to Z; the class of the loop t↦ e^2π it is a generator of it. For ever… 020N
Proposition 7 The mapping x↦ φ (e^2π ix) of R onto P₁(R) makes (R,0) a universal covering of (P₁(R), φ (1)). Let c:I→ S₁ be the path t↦ e^π it; the class of φ (c) is a generator of the group π₁(… 020O
Lemma 1 Let X be a topological space, let G be a discrete group operating properly in X, and let p be the canonical projection of X onto X/G. 020P
Proposition 8 Let X be a topological space and let G be a discrete group operating properly in X. Let us denote by p the canonical projection of X onto X/G. 020Q
Remark 1 Theorem 4 of III, p. 287 and Prop. 8 (IV, p. 349) remain valid under the more general assumption that G is a finite-dimensional Lie group over R operating properly in X. For more d… 020R

§ 2. Poincaré groups of unravelable spaces

Proposition 1 Let A be an unravelable topological space, let B be a topological space locally connected by arcs, and let f: A→ B be a continuous mapping. Let a be a point of A and b=f(a). Suppos… 020S
Proposition 2 Let A be a connected topological space locally connected by arcs, let B be an unravelable topological space, and let f: A→ B be a continuous mapping. Let a be a point of A and b= f… 020T
Proposition 3 Let B be an unrollable topological space and let A be a connected and locally arcwise connected subspace of B (for example an open and connected subset). Let a be a point of A. The… 020U
Definition 1 Let X and Y be topological spaces and let f be a continuous mapping of X into Y. One says that the mapping f is relatively connected if every point of Y possesses a fundamental sys… 020V
Proposition 4 Let X and Y be topological spaces and let f: X→ Y be a relatively connected continuous mapping. 020W
Corollary 1 Let X and Y be topological spaces and let f: X→ Y be a continuous and relatively connected mapping. By passing to connected components, the mapping f induces a bijection from the s… 020X
Corollary 2 Let X and Y be topological spaces and let f: X→ Y be a continuous mapping. In order that f be relatively connected, it is necessary and sufficient that the following three properti… 020Y
Corollary 3 Let X and Y be topological spaces and let f: X→ Y be a continuous and relatively connected mapping. Let F be a set and let g: X→ F be a locally constant mapping. There exists a uni… 020Z
Example 1 Let X be a topological space and let R be an equivalence relation in X. Let Y denote the quotient topological space X/R and f: X→ Y the canonical mapping. Suppose that the equivale… 0210
Example 2 Let Y be a topological space locally connected by arcs, and let X be an open subset of Y. The space C_c(I; X) of paths in X is identified with a subspace of the space C_c(I; Y) of… 0211
Proposition 5 Let X and Y be topological spaces and let f: X→ Y be a continuous and relatively connected mapping. For every pair (T,T') of coverings of Y, the mapping f^*:C_Y(T; T')→ C_X(X×_YT… 0212
Corollary 1 Let X and Y be topological spaces and let f: X→ Y be a continuous relatively connected mapping. Let T and T' be coverings of Y. If the coverings X×_YT and X×_YT' of X are isomorphi… 0213
Corollary 2 Let X and Y be topological spaces and let f: X→ Y be a continuous and relatively connected mapping. If the space X is simply connected, the same is true of the space Y. 0214
Corollary 3 Let X be a topological space locally connected by arcs, Y an unravelable topological space, and let f: X→ Y be a continuous relatively connected mapping. For every point x of X, th… 0215
Proposition 6 Let X and Y be topological spaces and let f: X→ Y be a continuous and relatively connected mapping. Suppose that every point of Y possesses an open neighbourhood V whose inverse im… 0216
Corollary Let X and Y be contractible topological spaces and let f: X→ Y be a continuous and relatively connected mapping. Suppose that every point of Y possesses a neighbourhood V whose inv… 0217
Remark Let Y be a topological space, and let X, X' and Y' be subspaces of Y such that X⊂ X'⊂ Y'⊂ Y. Suppose that the canonical injection of X into Y is relatively connected. For every con… 0218
Example Let Y be a locally finite-dimensional differentiable manifold and let Z be a closed submanifold of Y (VAR, R, 5.8.3). Put X = Y-Z. 0219
Theorem 1 Let X be a compact contractible topological space and let x be a point of X. The Poincaré group π₁(X, x) has a finite presentation. 021A
Lemma 1 Let X be a topological space and let A be a subset of X. In order that A be meagre, it is necessary and sufficient that there exist an open covering (Uᵢ)_i∈ I of X such that A∩ Uᵢ… 021B
Lemma 2 Let X be a topological space and let A be a subset of X. 021C
Remark 1 Let G be a topological group and let A be a meagre subset of G. Suppose that A contains a nonempty open subset U of G and let y be a point of U. Then U is meagre and for every x∈ G… 021D
Proposition 7 Let X be a separated space. Every Souslinian subspace (TG, IX, p. 59, def. 2) of X is approachable. 021E
Lemma 3 Let (Uᵢ)_i∈ I be a finite family of nonempty open subsets of a topological space X and let O be an open dense subset of X× X. There exists a family (Vᵢ)_i∈ I of nonempty open subse… 021F
Proposition 8 Let P be a nonempty Polish topological space and let R be an equivalence relation in P whose graph is a meagre subset of P× P. There exists a continuous injective mapping of the se… 021G
Proposition 9 Let X be an unravelable topological space and let W be a basis for the topology of X. For every point x of X, the cardinal of the group π₁(X, x) is bounded above by sup(Card(W), Ca… 021H
Lemma 4 Let X be a Baire space, let G be a topological group acting continuously on X and let B be an approachable subset of X which is not meagre. The set of points g∈ G such that B∩ gB=n… 021I
Theorem 2 (Shelah[^1]) Let X be a connected Polish space which is locally connected by arcs, and let x be a point of X. If X is not unravelable, the group π₁(X, x) has the power of the continuum. 021J
Lemma 5 The set Γ is a meagre subset of K× K. 021K
Example Let P be the topological space which is the union of the circles with center (2/n,0) passing through the origin of the plane R², for ngeqslant 1 (III, p. 336, Exercise 6). The Poin… 021L

§ 3. Poincaré groups of topological groups

Definition 1 Given a topological group G and a group G', a local homomorphism of G into G' means a mapping f of a neighbourhood V of the identity element of G into G' such that, for every pair… 021M
Proposition 1 Let G and G' be topological groups and let p: G→ G' be a group homomorphism. In order that p should make G a covering of G', it is necessary and sufficient that the restriction of… 021N
Proposition 2 Let G be a connected topological group, let G' be a group and let f: V→ G' be a local homomorphism of G into G', where V is a connected neighbourhood of the identity element of G… 021O
Lemma 1 Let G be a topological group and let V be a connected neighbourhood of the identity element e of G. For every neighbourhood U of e in G and every x∈ V, there exist an integer n∈ N… 021P
Corollary 1 Let G be a simply connected topological group and let G' be a group. Let V be a connected neighbourhood of the identity element in G and let f: V→ G' be a local homomorphism. There… 021Q
Remark 1 When G =R, this corollary follows from proposition 6 of TG, V, p. 3. 021R
Corollary 2 Two locally connected and simply connected topological groups which are locally isomorphic are isomorphic. 021S
Corollary 3 Let G be a simply connected topological group and let V be a connected neighbourhood of the identity element of G. One defines a presentation of G by taking as generator set the se… 021T
Definition 2 A pointed topological space (X, e) endowed with a continuous law of composition m: X× X→ X is called a Hopf space, such that 021U
Example 1 Let G be a topological group and m its composition law. The identity element e of G is an identity element up to homotopy and (G, e) is a Hopf space. 021V
Example 2 Let X be a topological space and x a point of X. Endowed with the juxtaposition of loops at x, the pointed space (Ωₓ(X), eₓ) is a Hopf space. In fact, first one has eₓ*eₓ=eₓ. On th… 021W
Proposition 3 Let (X, e) be a Hopf space and m: X× X→ X its composition law. For any loops c,c' of X at e, one has: 021X
Remark Let (X, e) be a Hopf space. By Prop. 3, the mapping of composition of path classes, π₁(X, e)× π₁(X, e)→ π₁(X, e) is continuous if π₁(X, e) is endowed with the quotient topology of… 021Y
Proposition 4 Let (X, e) be a Hopf space (IV, p. 373, definition 2) and let m: X× X→ X be its law of composition. Assume that the space X is connected and locally arcwise connected. Let X' be a… 021Z
Proposition 5 Let us keep the notation and the hypotheses of Proposition 4. 0220
Corollary Let G be a connected topological group and locally connected by arcs. Let G' be a connected covering of G, let p be its projection and let e' be an element of the fibre N of the id… 0221
Proposition 6 Let G be a connected and coverable topological group. There exist a topological group widetildeG, with identity element widetildee, and a continuous homomorphism p:widetildeG→ G sa… 0222
Example Proposition 6 applies in particular when G is a connected Lie group over R or C. There then exists on widetildeG a unique structure of analytic manifold such that the projection p… 0223
Scholium Let G be a connected and unknottable topological group, let e be its identity element. Let widetildeG be a topological group simply connected by arcs, with identity element widetil… 0224
Proposition 7 In order that two connected unrollable topological groups be locally isomorphic, it is necessary and sufficient that their universal coverings be isomorphic topological groups. 0225
Proposition 8 Let G be a connected unrollable topological group and let widetildeG be a universal covering of G. Let V be a connected open neighbourhood of the identity element e of G such that… 0226

§ 4. Theory of descent

Definition 1 A descent data relative to f on the X-space (Z, p) is called a continuous mapping τ : Z×_YX→ Z satisfying the following two properties: 0227
Example 1 Let X and Y be topological spaces, let f: X→ Y be a continuous mapping and let (T, q) be a Y-space. Put Z = X×_YT. The mapping τ from Z×_YX into Z which associates with ((x, t), x'… 0228
Example 2 Let Y be a topological space, (Vᵢ)_i∈ I a family of subsets of Y, and for each i∈ I, let (Zᵢ, pᵢ) be a Vᵢ-space. Let X denote the sum topological space of the family (Vᵢ)_i∈ I and… 0229
Example 1 Let X and Y be topological spaces and f: X→ Y a continuous mapping. Let (T, q) be a Y-space. Let Z be the X-space X×_YT, endowed with the mapping pr₁; let us denote by τ its canoni… 022A
Example 2 Let us take up again the notation of Example 2 (IV, p. 383). The topological space Z/R_τ is then the topological space obtained by gluing the spaces Zᵢ along the overset-1pᵢ(Vᵢ∩ Vⱼ… 022B
Proposition 1 Let X and Y be topological spaces, let f: X→ Y be a continuous mapping. Suppose that every point of Y possesses a neighbourhood above which there exists a continuous section of the… 022C
Proposition 2 Let f: X→ Y be a continuous mapping, let (Z, p) be an X-space and let τ be a descent datum relative to f on Z. The equivalence relation R_τ is closed if f is proper; it is open if… 022D
Proposition 3 Let X and Y be topological spaces and let f: X→ Y be a continuous mapping. Let (Z, p) and (Z', p') be X-spaces endowed with descent data relative to f, denoted respectively by τ an… 022E
Proposition 4 Suppose that the mapping f is strict and surjective and that T is a spread Y-space. Then the mapping f^*:C_Y(T; T')→ C_f(X×_YT; X×_YT') is bijective. 022F
Corollary Let X and Y be topological spaces and let f: X→ Y be a strict and surjective mapping. Let T and T' be spread Y-spaces. If there exists an X-isomorphism X×_YT→ X×_YT' compatible wit… 022G
Proposition 5 Let X and Y be topological spaces and let f: X→ Y be a continuous mapping. Suppose that the mapping f is proper and separated, or else open. Every descent datum relative to f on a… 022H
Lemma 1 Let B, B' be topological spaces, and let f: B'→ B be a continuous mapping. If the fibred square B'×_BB' is locally connected, the topological space B' is locally connected. 022I
Proposition 6 Let 022J
Corollary 1 Let E and B be topological spaces, p: E→ B a continuous mapping and (Aᵢ)_i∈ I a locally finite closed covering of B such that for every pair (i, j)∈ I× I,i=not j, the intersection… 022K
Corollary 2 Let X and Y be topological spaces and let f: X→ Y be a proper, separated and surjective mapping. Assume in addition one of the following hypotheses: 022L
Proposition 7 Let X and Y be topological spaces and let f: X→ Y be a continuous and surjective mapping. Suppose that the space Y is unloopable and that the mapping f is open and has the path-lif… 022M
Remark Let X and Y be topological spaces and let f: X→ Y be a mapping. In order that every descent datum relative to f on a covering of X be effective, and that the quotient space be a co… 022N
Proposition 8 Let X and Y be non-empty topological spaces and let f be a continuous mapping of X into Y. Assume that the space X is locally arcwise connected, that the space Y is connected, and… 022O
Proposition 9 Let X and Y be topological spaces and let f be a continuous mapping of X into Y. Suppose that the space X is locally arcwise connected, that the space Y is simply connected, and th… 022P
Proposition 10 Let X and Y be topological spaces and let f be a continuous mapping of X into Y. Suppose that the space X is simply connected, that the space X×_YX is locally arcwise connected, an… 022Q
Lemma 2 Let us retain the notation and hypotheses of the proposition. Let T be a set endowed with an operation of the groupoid Coeg(f) relative to a mapping q: T→ Y. Then there exists a un… 022R
Theorem 1 Let X and Y be topological spaces, and let f: X→ Y be a continuous surjective mapping. Suppose that the spaces X and Y are locally path-connected and semi-locally simply connected… 022S
Example Here are two examples in which the hypotheses of theorem 1 are satisfied. 022T
Theorem 2 Suppose that every point of Y possesses a neighbourhood above which there exists a continuous section of the mapping f. The groupoid morphism varpi '(f) from Coeg(f) into varpi (Y)… 022U
Lemma 3 Let G be a groupoid and let p, q be integers geqslant 1. For every pair (j, k) of integers such that 0leqslant jleqslant p and 0leqslant kleqslant q, let x_j,k be a vertex of G; fo… 022V
Example 1 The theorem applies when the mapping f is étale and surjective. 022W
Example 2 It also applies when the space X is the sum space of a family (Vᵢ)_i∈ I of parts of Y whose interiors cover Y, and when f is the mapping induced by the canonical injections of each… 022X
Theorem 3 Let X be a delaceable topological space and let G be a discrete group operating properly in X; let f: X→ X/G be the canonical surjection. The canonical morphism of groupoids varpi… 022Y
Lemma 4 For every path c= (c₁, c₂) in X×_YX, we have β ([c₁]) =β ([c₂]). 022Z

§ 5. Van kampen theorem

Definition 1 One says that the mapping f satisfies the property (VK) if it is strict, surjective and if the morphism varpi '(f) is an isomorphism. 0230
Example 1 This property is satisfied under one of the following assumptions: 0231
Theorem 1 Suppose that Y is path connected and that f verifies the property (VK). With the preceding notations, there exists a unique group homomorphism 0232
Lemma 1 The quiver Γ identifies with a subquiver of Γ; the quivers Γ and Γ are connected. 0233
Proposition 1 Let Y be a pathwise connected topological space and let (Aᵢ)_i∈ I be a covering of Y by non-empty, pathwise connected subsets, indexed by a totally ordered set I. Suppose that the… 0234
Corollary 1 Under the assumptions of proposition 1, suppose moreover that, for every i∈ I, the image of the homomorphism from π₁(Aᵢ, a(i)) into π₁(Y, a(i)), induced by the canonical injection… 0235
Corollary 2 Under the assumptions of proposition 1, suppose moreover that for every i∈ I, the group π₁(Aᵢ, a(i)) is reduced to the identity element and that, for every triple (i₁, i₂, i₃) of p… 0236
Example 1 For the case of a covering formed of two sets, see No. 3. 0237
Example 2 Let G be a graph (II, p. 155, definition 1); let S denote the set of vertices of G, A the set of its oriented edges, o and t the origin and term mappings of A into S; for every ori… 0238
Corollary 3 Under the hypotheses of Prop. 1, suppose in addition that the framework of the covering (Aᵢ)_i∈ I is an oriented tree. The homomorphism N:ᵢ*_∈ Iπ₁(Aᵢ, a(i))→ π₁(Y, a(i₀)) deduced f… 0239
Example 3 (Plane deprived of $n$ points) The fundamental group of R² 0 is isomorphic to Z, and the class of the path t↦ e^2π it is a generator of it (IV, p. 347, corollary). More generally, let n be a natural number and l… 023A
Corollary 4 Under the hypotheses of Proposition 1, suppose that there exists a subset A of Y, connected by arcs and nonempty, such that the intersection Aᵢ∩ A_i' is equal to A for every pair (… 023B
Example 4 Let ((Xᵢ, xᵢ))_i∈ I be a family of pointed topological spaces. The bouquet of the family ((Xᵢ, xᵢ))_i∈ I is by definition, and is denoted by bigvee_i∈ I(Xᵢ, xᵢ), the quotient topol… 023C
Proposition 2 There exists a unique group homomorphism 023D
Example For ngeqslant 1, the sphere Sₙ is the union of two closed hemispheres, homeomorphic to the closed ball Bₙ₋₁ (TG, VI, p. 12), and whose intersection is identified with the sphere Sₙ… 023E
Corollary 1 The homomorphism μ is injective. More precisely, there exists a retraction associated with μ which is a group homomorphism. 023F
Corollary 2 If the group π₁(X, a) is trivial, the set A = B∩ C is arcwise connected; if it is commutative, the set A has at most two arcwise connected components. 023G
Corollary 3 If the groups π₁(X, a) and π₁(A, a) are trivial, the same is true of the groups π₁(B, a) and π₁(C, a). 023H
Proposition 3 Suppose that X is unlaceable. The homomorphism N is then surjective, and its kernel is the smallest distinguished subgroup of π₁(X, o)*F(G) containing the elements 023I
Corollary 1 Suppose that X is unlaceable and that the group G is generated by the stabilizers of the points of X. The canonical morphism π₁(f, o):π₁(X, o)→ π₁(Y, f(o)) is then surjective. In p… 023J
Remark If X is unlaceable, so is Y (IV, p. 349, prop. 8). Since a connected unlaceable space is simply connected by arcs if and only if it is simply connected (IV, p. 344, corollary 1 of… 023K
Example 1 Let X be an arcwise connected, unlaceable, separated topological space, and let a be a point of X. Let n be an integer geqslant 2 and let Y be the quotient of the space Xⁿ by the a… 023L
Corollary 2 Suppose that X is unlaceable and that the group G operates freely in X. There exists a unique group homomorphism p:π₁(Y, f(o))→ G whose kernel contains the image of π₁(X, o) and su… 023M
Corollary 3 Suppose that X is simply connected by arcs. The mapping of G into π₁(Y, f(o)) which, to g∈ G, associates the path class f_*(β_g) is a surjective group homomorphism; its kernel is t… 023N
Proposition 4 Suppose that the arcwise connected components of Y are open. For each i∈ I, let Gᵢ be the quotient of the group π₁(Yᵢ, bᵢ) by the smallest normal subgroup containing the image of t… 023O
Corollary 1 Suppose moreover that the spaces X and Y are arcwise connected and let a be a point of X. The canonical mapping of π₁(Y, f(a)) into π₁(Cone(f), f(a)) is surjective, and its kernel… 023P
Corollary 2 Suppose moreover that the arcwise connected components of Y are simply arcwise connected. The homomorphism μ: F(J-T)→ π₁(Cone(f), s) is then an isomorphism. 023Q
Remark 1 Let X be a topological space whose arcwise connected components are open. Let A be a closed subspace of X; let ι : A→ X denote the canonical injection, let X/A denote the space ded… 023R
Lemma 2 The mapping f: C→ A is proper and separated; its fibres are finite. 023S
Proposition 5 (Van Kampen) The homomorphism Q is surjective; its kernel is the smallest normal subgroup containing the elements 023T
Remark Let A be a topological space, let B be a closed subset of A, and let U be an open neighbourhood of B in A. Suppose that the set U-B is the union of a finite family (M_ℓ)_ℓ∈ L of pa… 023U

§ 6. Classifying spaces

Proposition 1 Let X' be a normal topological space, let X be a subspace of X', let A be a closed subspace of X' contained in X, and let U be a neighbourhood of A in X. 023V
Corollary 1 Let X be a normal topological space, let f be a continuous mapping of X into a topological space Z. Let A be a closed subspace of X, let U be a neighbourhood of A in X and let σ… 023W
Corollary 2 Let X be a normal topological space, let A be a closed subspace of X and let U be a neighbourhood of A in X. Let f and f' be continuous homotopic mappings of U into a topological s… 023X
Corollary 3 Let X be a normal topological space, let A be a closed subspace of X and let U be a neighbourhood of A in X. Let Z be a topological space homotopic to a point and let g: U→ Z be a… 023Y
Corollary 4 Let X be a paracompact topological space, let A be a closed subspace of X. Let n be an integer geqslant 0 and let V be an open subset of Rⁿ. Let f: X→ V be a continuous mapping and… 023Z
Lemma 1 Let Z be a topological space, K a compact topological space and W an open subset of Z× K. The set U of points z∈ Z such that z × K is contained in W is open in Z. 0240
Corollary 5 Let X' be a normal topological space, let X be a subspace of X', let A be a closed subspace of X' contained in X, and let U be a neighbourhood of A in X. 0241
Proposition 2 Let B be a paracompact topological space and let (E, p) be a locally trivial fibre space over B× I. Put E₁=overset-1p(B× 1) and denote by p₁: E₁→ B the mapping pr₁∘ p|E₁. Then the… 0242
Lemma 2 Let α ,β ,γ be real numbers such that α leqslant β leqslant γ, let B be a topological space and let p: E→ B× [α , γ ] be a continuous mapping. Put B₀= B× [α , β ], B₁= B× [β , γ ]… 0243
Lemma 3 Let B be a topological space and let (E, p) be a locally trivial B× I-fibered space. Every point a of B possesses a neighbourhood V such that the V× I-space E_V×I is trivializable. 0244
Corollary 1 Let B be a paracompact topological space and let (E, p) be a locally trivial fibre space over B× I (I, p. 71, Corollary 2). For t∈ I, let us denote by (Eₜ, pₜ) the locally trivial… 0245
Corollary 2 Let B and B' be topological spaces, and let E be a locally trivial fibre space over B. Let f₀ and f₁ be continuous mappings of B' into B. Suppose that the space B' is paracompact… 0246
Corollary 3 Let B be a paracompact topological space. If B is homotopic to a point, every locally trivial fibre space with base B is trivializable. 0247
Corollary 4 Let B and B' be topological spaces, let (E, p) be a locally trivial fibre space over B, let f be a continuous mapping of B' into B, let σ : B'× I→ B be a homotopy starting from f… 0248
Lemma 4 Let B be a topological space, let G be a topological group and let E, E' be principal fibre spaces with group G and base B. Let F denote the topological space G endowed with the le… 0249
Lemma 5 The morphism of sheaves α is an isomorphism. 024A
Proposition 3 Let G be a topological group, let B be a paracompact topological space and let (E, p) be a principal fibre space of group G and base B× I. Put E₁=overset-1p(B× 1) and let p₁: E₀→ B… 024B
Corollary 1 Let B be a paracompact topological space, let G be a topological group and let (E, p) be a principal fibre B× I-space of group G. For t∈ I, let (Eₜ, pₜ) denote the principal fibre… 024C
Corollary 2 Let B and B' be topological spaces, let G be a topological group, and let E be a B-principal fibre space with group G. Let f₀ and f₁ be continuous mappings of B' into B. Suppose th… 024D
Corollary 3 Let B be a paracompact topological space and let G be a topological group. If B is homotopic to a point, every principal fibre space with group G is trivialisable. 024E
Remark An alternative proof of these results would consist in verifying that the isomorphisms of fibre spaces constructed in prop. 2 and its corollaries are isomorphisms of principal fibr… 024F
Proposition 4 If the space B is paracompact and if the space E' is homotopic to a point, the sheaf on B of morphisms of principal fibre spaces with group G from E into E' is a soft sheaf. 024G
Theorem 1 Let G be a topological group, let Bᵤ be a topological space, and let (Eᵤ, pᵤ) be a principal fibre space with group G and base Bᵤ. Suppose that the space Eᵤ is homotopic to a point… 024H
Corollary Let G be a topological group, let B and B' be paracompact topological spaces. Let (E, p) and (E', p') be principal fibre spaces with group G and with base B and B' respectively. Su… 024I
Example Endowed with the mapping p:R→ S¹,t↦ e^2π it, and with the operation of Z by translation, the space R is a principal covering space with group Z. The space S¹ is thus a classifying… 024J
Proposition 5 There exists a unique topology on G^* compatible with its group structure for which the sets V^*_ε form a basis of neighbourhoods of the identity element. 024K
Proposition 6 The space G^* is contractible and locally contractible at each of its points. 024L
Remark 1 If G is a separated topological group, ι (G) is closed in G^*. 024M
Proposition 7 Suppose that G is a metrizable topological group. Then the topological group G^* is metrizable. More precisely, if d is a bounded distance on G which defines its topology, the topo… 024N
Remark 2 Let d be a bounded distance which defines the topology of G. When the topological group G^* is endowed with the distance d^*, the homomorphism ι : G→ G^* is an isometry. 024O
Remark 3 Suppose that G is a discrete topological group. The subgroup ι (G) is then a discrete subgroup of G^*. In fact, the topology of G is defined by the distance d on G given by d(g, g'… 024P
Theorem 2 Let G be a discrete topological group. Let G operate on the right in G^* by h· g=hι (g) and let B denote the quotient topological space G^*/G. 024Q