It is sufficient to prove that, if A and B are two unloopable spaces, the same is true of their product $A\times B$. Under these conditions, the space $A\times B$ is in fact locally arcwise connected (III, p. 261, prop. 9). Let $(a, b)$ be a point of $A\times B$. By hypothesis there exists a neighbourhood U of $a$ (resp. a neighbourhood V of $b$) such that the image of the homomorphism $i_*:\pi_1(U, a)\rightarrow \pi_1(A, a)$ deduced from the canonical injection $i: U\rightarrow A$ (resp. of the homomorphism $j_*:\pi_1(V, b)\rightarrow \pi_1(B, b)$ deduced from the canonical injection $j: V\rightarrow B$) is reduced to the identity element. Then $U\times V$ is a neighbourhood of $(a, b)$ in $A\times B$. The homomorphism $\pi_1(U\times V,(a, b))\rightarrow \pi_1(A\times B,(a, b))$ identifies with the homomorphism $(i_*, j_*)$ (III, p. 297, corollary of prop. 4). Its image is therefore reduced to the identity element. This proves that $A\times B$ is unloopable.

b) Conversely, let $p: E\rightarrow B$ be an étale and surjective mapping and suppose that E is an unloopable space. Then B is unloopable.

a) Let B be an unloopable topological space and let $(E, p)$ be a covering of B. Then the space E is locally arcwise connected. Let $x$ be a point of E, put $b=p(x)$, and let V be a neighbourhood of $b$ in B such that the canonical homomorphism $\pi_1(V, b)\rightarrow \pi_1(B, b)$ is trivial. Let $i: V\rightarrow B$ and $j:\overset{-1}{p}(V)\rightarrow E$ denote the canonical injections. By hypothesis, the image of the homomorphism $\pi_1(i, b)$ is reduced to the identity element. Since $p\circ i=j\circ p$ and $\pi_1(p, x)$ is injective, the image of the homomorphism $\pi_1(j, x)$ is reduced to the identity element. This proves that E is unloopable.

b) The space B is locally arcwise connected. Let $b$ be a point of B and let $x$ be a point of $E_b$. Since $p$ is étale, there exists a neighbourhood U of $x$ in E such that $p$ induces a homeomorphism of U onto $p(U)$. Let V be a neighbourhood of $x$ in E contained in U such that the homomorphism $\pi_1(j, x)$ is trivial, where $j: V\rightarrow E$ denotes the canonical injection. Let $q: V\rightarrow p(V)$ be the mapping deduced from $p$ by passing to subspaces; it is a homeomorphism. Let also $i$ denote the canonical injection of $p(V)$ into B. We have $p\circ j=i\circ q$, hence $\pi_1(i, b)\circ \pi_1(q, x) =\pi_1(p, x)\circ \pi_1(j, x)$ is the trivial homomorphism. Since the homomorphism $\pi_1(q, x)$ is an isomorphism, the homomorphism $\pi_1(i, b)$ is trivial. It follows that the space B is unloopable.

In fact, the mapping $q\circ p$ is étale (I, p. 29, prop. 6) and separated (I, p. 25, prop. 2). By Remark 2 above, it is therefore enough to prove that it satisfies the path-lifting property. Let $z$ be a point of Z and let $c:\mathbf{I}\rightarrow B$ be a path in B with origin $q(p(z))$. There exists a path $c'$ with origin $p(z)$ in Y which lifts $c$, because Y is a covering of B (III, p. 302, Cor. 2). Since Z is a covering of Y, there exists a path $c''$ with origin $z$ in Z which lifts $c'$; the path $c''$ lifts $c$. This proves the proposition.

(i) The space B is unlaceable.

(ii) There exists a nonempty covering of B which is simply connected by arcs.

(iii) The space $\lambda_b(B)$, endowed with the endpoint mapping $\varepsilon_B:\lambda_b(B)\rightarrow B$, is a covering of B.

(iv) The group $\pi_1(B, b)$ is discrete for the admissible topology.

(v) The group $\pi_1(B, b)$ is discrete for the quotient topology of the compact-convergence topology.

Moreover, when these conditions are satisfied, the space $\lambda_b(B)$ is simply connected by arcs, simply connected, Galois with group $\pi_1(B, b)^{\circ}$, and the pointed covering $(\lambda_b(B), \varepsilon_b)$ is a universal covering of the pointed space $(B, b)$.

It follows from the definition of an unlaceable space that the subgroup of $\pi_1(B, b)$ reduced to the identity element is admissible if and only if B is unlaceable. This proves the equivalence (i)$\Leftrightarrow ($iv). Moreover, the equivalence of properties (iv) and (v) follows from Remarks 4 and 5 of III, p. 320.

(iv)$\Rightarrow ($ii). By III, p. 316, Prop. 5, there exists a covering $(E, p)$ of B and a point $x\in E_b$ such that $p_*(\pi_1(E, x)) =\{e\}$; in particular, $\pi_1(E, x)$ is reduced to the identity element. The connected component by arcs of $x$ in E is then a nonempty covering of B (I, p. 80, Cor. 1 to Prop. 6), simply connected by arcs.

(ii)$\Rightarrow ($iii). Let E be a nonempty covering of B which is simply connected by arcs. Let $x$ be a point of $E_b$ (there exists one because B is connected, I, p. 74, Prop. 4). The projection $q: E\rightarrow$ B induces a homeomorphism $\Lambda_x(E)\rightarrow \Lambda_b(B)$ (III, p. 302, Cor. 2 to Prop. 3), whence, by passing to connected components by arcs, a homeomorphism $q_*:\lambda_x(E)\simeq \lambda_b(B)$. The endpoint mapping $\Lambda_x(E)\rightarrow E$ is continuous and open, because E is locally connected by arcs (III, p. 264, Corollary). The mapping $\varepsilon_E:\lambda_x(E)\rightarrow E$ which it defines by passing to connected components by arcs is therefore continuous and open. The mapping $\varepsilon_E\circ (q_*)^{-1}:\lambda_b(B)\rightarrow E$ is bijective, continuous and open. This proves that the topological space $\lambda_b(B)$, endowed with the compact-convergence topology and the endpoint mapping, is a covering of B.

(iii)$\Rightarrow ($v). In fact, the set $\pi_1(B, b)$, endowed with the quotient topology of the compact-convergence topology, is identified with the fibre of the B-space $\lambda_b(B)$ over the class of the constant loop at $b$. If $\lambda_b(B)$ is a covering of B, then $\pi_1(B, b)$ is discrete for the compact-convergence topology.

Suppose that these assertions are verified. From the foregoing, the space $\lambda_b(B)$ is then a covering of B which is simply connected by arcs, hence simply connected. The pointed space $(\lambda_b(B), \varepsilon_b)$ is consequently a universal covering of $(B, b)$ (I, p. 126, corollary to Prop. 3) and the covering $\lambda_b(B)$ is a Galois covering of B (I, p. 120, Prop. 1). The canonical homomorphism $h_{(\lambda_b(B),\varepsilon_b)}:\pi_1(B, b)\rightarrow$ Aut$_B(\lambda_b(B))$ is then an isomorphism (III, p. 306, Prop. 5).

(i) The group $\pi_1(E, x)$ is trivial.

(ii) The space E is simply connected.

(iii) The pointed space $(E, x)$ is a universal covering of $(B, b)$.

The implication (i)$\Rightarrow$(ii) has already been proved under the sole assumption that the space B is connected and locally arcwise connected (III, p. 309, cor. 2 of prop. 1), and the implication (ii)$\Rightarrow$(iii) without any assumption (I, p. 126, corollary).

Let us prove (iii)$\Rightarrow$(i). By Theorem 1 of IV, p. 342, there exists a covering $E'$ of B and a point $x'$ of the fibre $E'_b$ such that the group $\pi_1(E', x')$ is reduced to the identity element. Under assumption (iii), there exists a pointed B-morphism $f: (E, x)\rightarrow (E', x')$. Let $p: E\rightarrow$ B and $p': E'\rightarrow B$ denote the projections of the coverings E and $E'$; we have $p=p'\circ f$, hence $\pi_1(p, x) =\pi_1(p', x')\circ \pi_1(f, x)$. Since the group $\pi_1(E', x')$ is trivial, the injective homomorphism $\pi_1(p, x)$ has the identity element for image. This proves that the group $\pi_1(E, x)$ is reduced to the identity element.

By prop. 7 of I, p. 81, the space E, endowed with the mapping $f$, is a covering of $E'$. The last assertion then follows from Corollary 1.

Let B be an unwindable topological space and let $a,b$ be two points of B. The quotient topological space $\varpi_{a,b}(B)$ is homeomorphic to the space $\pi_1(B, a)$, endowed with the quotient topology of the compact-open topology (III, p. 293, remark 3), and therefore is discrete by Theorem 1 of IV, p. 342. Consequently, the strict homotopy class of every path in B joining $a$ to $b$ is an open subset of $\Lambda_{a,b}(B)$. More generally:

Suppose first that the space B is simply arcwise connected. Let $\varphi : \Lambda (B)\rightarrow B\times B$ denote the mapping which associates with a path $c$ in B the pair $(c(0), c(1))$ formed by its origin and its end. In order that two paths in B be strictly homotopic, it is necessary and sufficient that they have the same origin and the same end (III, p. 292, prop. 2). Since the space B is connected and locally arcwise connected, the mapping $\varphi$ is surjective, continuous and open (III, p. 262, prop. 10). The relation of strict homotopy is therefore open (TG, I, p. 32, prop. 3).

In the general case, let E be a simply connected by paths covering of B, nonempty if B $=\not\emptyset$. This covering is surjective because B is connected (I, p. 74, prop. 4). Let us denote by $p$ the projection of the B-space E and let $\widetilde{p}: \Lambda (E)\rightarrow \Lambda (B)$ be the mapping which associates to a path $c$ in E the path $p\circ c$. Endowed with the mapping $\widetilde{p}$, Λ(E) is a covering of Λ(B) (III, p. 302, cor. 1 of prop. 3). In order that two paths in B should be strictly homotopic, it is necessary and sufficient that they should be the images under the mapping $\widetilde{p}$ of two strictly homotopic paths in E (III, p. 302, prop. 4). Let U be an open subset of the space Λ(B). The set $(\overset{-1}{\widetilde{p}})(U)$ is open in Λ(E) ; by the first part of the

proof, the set $U'$ saturated of $(\overset{-1}{\widetilde{p}})(U)$ for the relation of strict homotopy is open in Λ(E). Since Λ(E) is a covering of Λ(B), the set $\widetilde{p}(U')$ is open in Λ(B). This proves the proposition, for $\widetilde{p}(U')$ is the saturated set of U for the relation of strict homotopy.

Let B be a locally contractible space. Let $b$ be a point of B and let V be a neighbourhood of $b$ such that $(V, b)$ is a contractible pointed space. The space V is homotopic to a point, hence $\pi_1(V, b) =\{\varepsilon_b\}($III, p. 296, corollary 3).

To prove the proposition, it remains to prove that the space B is locally connected by paths. Let $\sigma : V\times \mathbf{I}\rightarrow$ V be a pointed homotopy at $b$ joining the constant mapping of image $b$ to the mapping Id$_V$. For every neighbourhood W of $b$ contained in V, the set $\sigma (W\times \mathbf{I})$ is a neighbourhood of $b$ because it contains $W =\sigma (W\times \{1\})$, and it is connected by paths because for every $(a, s)\in W\times \mathbf{I}$, the mapping $t\mapsto \sigma (a, ts)$ is a path joining $b=\sigma (a,0)$ to $\sigma (a, s)$ in $\sigma (W\times \mathbf{I})$. Let us prove that the sets of the form $\sigma (W\times \mathbf{I})$ form a fundamental system of neighbourhoods of $b$. Let $V'$ be an open neighbourhood of $b$ in B; then, $\overset{-1}{\sigma}(V')$ is an open neighbourhood of $\{b\} \times \mathbf{I}$ in $V\times \mathbf{I}$. Since the space $\mathbf{I}$ is compact, the projection pr$_1: V\times \mathbf{I}\rightarrow V$ is proper (TG, I, p. 77, cor. 5) and pr$_1(\complement \overset{-1}{\sigma}(V'))$ is a closed subset of V not containing $b$. Its complement W is thus an open neighbourhood of $b$ in V such that $W\times \mathbf{I}\subset \overset{-1}{\sigma}(V')$, whence $\sigma (W\times \mathbf{I})\subset V'$.

Every open subset of a numerical space or of a projective space, real or complex, of dimension $n$ (cf. Chapters VI and VIII) is unlaceable, as are the Euclidean spheres $\mathbf{S}_n$ (TG, VI, p. 11, prop. 4). More generally, every topological manifold (VAR R, second part, p. 7, Notations et Conventions) is unlaceable. In fact, these spaces are locally contractible.

2) Poincaré group of the circle. — The topological space $\mathbf{R}$ is homotopic to a point (III, p. 234, example 4), hence simply connected by paths (IV, p. 340). The canonical surjection $p$ of $\mathbf{R}$ onto $\mathbf{T}=\mathbf{R}/\mathbf{Z}$ makes it a principal covering of group $\mathbf{Z}($I, p. 100, example 4). The topological space $\mathbf{T}$ is unlaceable and $(\mathbf{R},0)$ is a universal covering of $(\mathbf{T},0)$. One deduces from this a canonical isomorphism of groups $\pi_1(\mathbf{T},0)\rightarrow \mathbf{Z}:$ it maps the class of the loop $t\mapsto p(nt)$ at 0 to the element $n$ of $\mathbf{Z}$. Taking into account example 6 (I, p. 101), one deduces the following proposition.

Only the last assertion remains to be proved. The fibre $E_1$ over 1 of a connected nonempty covering E of $\mathbf{S}_1$ is endowed with a transitive operation of the group $\pi_1(§_1,1)$. Since the subgroups of $\mathbf{Z}$ are the sets $n\mathbf{Z}$, for $n\geqslant 0$, we see that $E_1$ is isomorphic to one of the homogeneous sets associated with the coverings described above. Since $\mathbf{S}_1$ is connected and locally arcwise connected, E is isomorphic to one of these coverings.

The mapping $z\mapsto (|z|, z/|z|)$ is a homeomorphism of the space $\mathbf{C}^*$ onto the space $\mathbf{R}^*_+\times \mathbf{S}^1$ (TG, VI, p. 10, prop. 3); in particular, $\mathbf{C}^*$ is arcwise connected. Since the mapping $x\mapsto e^x$ of $\mathbf{R}$ into $\mathbf{R}_+^*$ is a homeomorphism, it follows from proposition 6 that the mapping $(x, y)\mapsto e^xe^{2\pi iy}$ makes $\mathbf{R}^2$ a covering of $\mathbf{C}^*$. On identifying $\mathbf{R}^2$ with $\mathbf{C}$ by means of the mapping $(x, y)\mapsto x+iy$, we conclude that the space $\mathbf{C}$, endowed with the mapping $z\mapsto e^z$, is a covering of $\mathbf{C}^*$. Since the space $\mathbf{C}$ is connected and simply arcwise connected, the pointed covering $(\mathbf{C},0)$ is a universal covering of the pointed space $(\mathbf{C}^*,1)$.

It also follows from the corollary, III, p. 297, that the Poincaré group of $\mathbf{C}^*$ at 1 is isomorphic to $\mathbf{Z}$ and that the class $\gamma$ of the loop $t\mapsto e^{2\pi it}$ is a generator of it.

Let $n$ be an integer $>0$. The mapping $x\mapsto x^n$ of $\mathbf{R}^*_+$ onto itself is a homeomorphism; it follows as above that the mapping $z\mapsto z^n$ makes $\mathbf{C}^*$ a covering of degree $n$ of $\mathbf{C}^*$. It is Galoisian with group $\mathbf{Z}/n\mathbf{Z}$. The last assertion is proved as in the proof of prop. 6.

For every integer $n\geqslant 0$, the torus $(\mathbf{S}_1)^n$ is an unwindable space (IV, p. 341, prop. 1) and its Poincaré group at every point is isomorphic to $\mathbf{Z}^n($III, p. 297, prop. 4).

We shall generalize these results in paragraph 3 devoted to coverings of topological groups.

3) Real projective spaces. --- Let $n$ be an integer $>0$. The spaces $\mathbf{S}_n$ and $\mathbf{P}_n(\mathbf{R})$ are connected, non-empty, and the canonical mapping (TG, VI, p. $13$)$\varphi :\mathbf{S}_n\rightarrow \mathbf{P}_n(\mathbf{R})$ makes $\mathbf{S}_n$ a principal covering of $\mathbf{P}_n(\mathbf{R})$ with group $\{+1,-1\}($I, p. 99, example 1). For $n\geqslant 2$, the sphere $\mathbf{S}_n$ is simply connected (I, p. 127, example 3) and unwindable, hence simply connected by arcs (IV, p. 344, corollary 1). For $n= 1$, the mapping $z\mapsto z^2$ of $\mathbf{S}_1$ into $\mathbf{S}_1$ defines an equivalence relation in $\mathbf{S}_1$ whose classes are the pairs of opposite points of $\mathbf{S}_1$. The mapping $\varphi :\mathbf{S}_1\rightarrow \mathbf{P}_1(\mathbf{R})$ defines by passing to the quotient a continuous bijection $\psi :\mathbf{S}_1\rightarrow \mathbf{P}_1(\mathbf{R})$ such that $\psi (z^2) =\varphi (z)$. Since $\mathbf{S}_1$ is a compact space, the mapping $\psi$ is a homeomorphism (TG, I, p. 63, cor. 2).

For every integer $n\geqslant$ 2 and every point $x$ of $\mathbf{S}_n$, the canonical mapping $\varphi :\mathbf{S}_n\rightarrow \mathbf{P}_n(\mathbf{R})$ makes $(\mathbf{S}_n, x)$ a universal covering of $(\mathbf{P}_n(\mathbf{R}), \varphi (x))$. For every path $c$ in $\mathbf{S}_n$ joining $x$ to $-x$, the class of $\varphi \circ c$ generates the group $\pi_1(\mathbf{P}_n(\mathbf{R}), \varphi (x))$, which is isomorphic to $\mathbf{Z}/2\mathbf{Z}$.

4) Quotient of a space by the proper action of a discrete group

Let $x$ be a point of X, let $K_x$ be its stabilizer in G. Let $U_1$ be a neighbourhood of $x$ in X; there exists a neighbourhood U of $x$ in X, stable under $K_x$ and contained in $U_1$ such that $g\cdot U\cap U =\emptyset$ if $g\notin K_x$ and such that the canonical mapping $p$ induces a homeomorphism of $U/K_x$ onto a neighbourhood V of $p(x)$ in $X/G$.

By prop. 8 of TG, III, p. 32, there exists a neighbourhood $U_2$ of $x$ in X, stable under $K_x$, such that $g\cdot U_2\cap U_2=\emptyset$ if $g\notin K_x$ and such that the canonical mapping $p$ induces a homeomorphism of $U_2/K_x$ onto a neighbourhood of $p(x)$ in $X/G$.

Since the canonical mapping $U_2\rightarrow U_2/K_x$ is closed (TG, III, p. 28, prop. 2), there exists an open neighbourhood U of $x$ contained in $U_1\cap U_2$ which is stable under $K_x($I, p. 75, lemma). The equivalence relation in $U_2$ defined by $K_x$ is also open (TG, III, p. 10, lemma 2). It then follows from TG, I, p. 32, prop. 4, that the canonical mapping induces a homeomorphism of $U/K_x$ onto an open neighbourhood of $p(x)$ in $X/G$, whence the lemma.

a) Suppose that X is connected by arcs and that the group G is generated by the stabilizers of the points of X. Then the canonical homomorphism $\pi_1(X, x)\rightarrow \pi_1(X/G, p(x))$ is surjective for every point $x$ of X. In particular, $X/G$ is simply connected by arcs if X is.

b) If the space X is unloopable, the space $X/G$ is unloopable.

a) The set N of the elements $g\in G$ for which there exists a path $c_g:\mathbf{I}\rightarrow X$ such that $c_g(0) =x,c_g(1) =g\cdot x$ and such that the loop $p\circ c_g$ in $X/G$ is strictly homotopic to a constant loop is a subgroup of G. If $g\in G$ and if there exists a point $y$ of X such that $g\cdot y=y$, let us choose a path $c:\mathbf{I}\rightarrow X$ joining $x$ to $y$; then the path $c'=c*$ $\overline{(g\cdot c)}$ satisfies $c'(0) =x,c'(1) =g\cdot x$ and $[p\circ c'] = [p\circ c][p\circ (g\cdot c)]^{-1}=e_{p(x)}$, whence $p\circ c'$ is strictly homotopic to a constant loop. Since G is generated by the stabilizers of the points of X, it follows that N = G.

Let $c$ be a loop in $X/G$ at $p(x)$. By Theorem 4 of III, p. 287, there exists a path $\widetilde{c}:\mathbf{I}\rightarrow X$ lifting $c$ such that $\widetilde{c}(0) =x$. Since $p(\widetilde{c}(1)) =c(1) =p(x)$, there exists $g\in G$ such that $\widetilde{c}(1) =g\cdot x$. Choose a path $c_g:\mathbf{I}\rightarrow X$ joining $x$ to $g\cdot x$ and such that $p\circ c_g$ is strictly homotopic to a constant loop. Then the path $c'=\widetilde{c}*\overline{c_g}$ is a loop at $x$ in X such that $[p\circ c'] = [c]$. This shows that the homomorphism $\pi_1(X, x)\rightarrow \pi_1(X/G, p(x))$ is surjective. The other assertion follows immediately.

Let us now prove assertion b). The space $X/G$ is locally arcwise connected (III, p. 261, Prop. 8).

Let $x$ be a point of X and let $K_x$ be its stabilizer in G. Let $U_1$ be an open neighbourhood of $x$, contained in U, such that the image of the canonical homomorphism $\pi_1(U_1, x)\rightarrow \pi_1(X, x)$ is reduced to the identity element. By Lemma 1 (IV, p. 349), there exists a neighbourhood U of $x$ in X, contained in $U_1$, stable under $K_x$, such that $g\cdot U\cap U =\emptyset$ if $g\notin K_x$ and such that the canonical mapping $p$ induces a homeomorphism of $U/K_x$ onto a neighbourhood V of $p(x)$ in $X/G$.

Let $c$ be a loop at $x$ in V; by Theorem 4 (III, p. 287), applied to the topological space U and the group $K_x$, there exists a path $\widetilde{c}:\mathbf{I}\rightarrow U$ such that $\widetilde{c}(0) =x$ and which lifts $c$. Necessarily $\widetilde{c}(1) =x$, so that $\widetilde{c}$ is a loop at $x$ in U. By assumption, $\widetilde{c}$ is strictly homotopic to a constant loop in X; consequently, the same is true of $c$, and the canonical homomorphism $\pi_1(V, p(x))\rightarrow \pi_1(X/G, p(x))$ is trivial. This proves that $X/G$ is unloopable if X is.

5) In the Euclidean plane $\mathbf{R}^2$, let P be the topological space equal to the union of the circles with center $(1/n,0)$ passing through the origin (for $n\in \mathbf{N}^*$). The space P is compact, connected and locally arcwise connected, but is not unloopable. The group $\pi_1(P,0)$, endowed with the admissible topology, is separated and non-discrete (III, p. 337, Exerc. 7).