Proposition 11

Let B be a topological space, G a topological group and $\mathscr{U}= (U_i)_{i\in I}$ an open covering of B.

a) Every principal fibre space over B with group G which is trivializable over $\mathscr{U}$ is isomorphic to a principal fibre space defined by a cocycle of $Z^1_{cont}(B,\mathscr{U},G)$.

b) Let $(E, p)$ and $(E', p')$ be principal fibre spaces over B which are trivializable over $\mathscr{U}$. Let $(f_i)_{i\in I}($ resp. $(f'_i)_{i\in I})$ be a trivialization of $(E, p)$ ( resp. of $(E', p')$) adapted to $\mathscr{U}$, and let us denote by $(g_{ij})_{(i,j)\in I\times I}$ ( resp. $(g'_{i,j})_{(i,j)\in I\times I}$) the cocycle defined by this trivialization. Then the principal fibre spaces $(E, p)$ and $(E', p')$ are isomorphic if and only if these cocycles are cohomologous.

Let us prove b). Let $\varphi : E\rightarrow E'$ be an isomorphism of principal fibre spaces with base B and group G. For each $i\in I$, let $h_i$ be the continuous mapping of $U_i$ into G defined by

Since, for each $i\in I,f_i$ and $f'_i$ are compatible with the operations of G, one has for every $b\in U_i$ and every $g\in G$,

and

Hence, for every pair $(i, j)\in I\times I$ and every point $b$ of $U_i\cap U_j:$ $f'_i\circ (f'_j)^{-1}(b, e) = (f'_i\circ \varphi \circ f_i^{-1})\circ (f_i\circ f_j^{-1})\circ (f_j\circ \varphi^{-1}\circ (f'_j)^{-1})(b, e)$

so that one has

This proves that the cocycles $(g_{ij})$ and $(g'_{ij})$ are cohomologous.

Conversely, suppose that these cocycles are cohomologous, and let $(h_i)_{i\in I}$ be a family of continuous mappings $h_i: U_i\rightarrow G$ such that $g'_{ij}(b) =h_i(b)g_{ij}(b)h_j(b)^{-1}$ for $i,j\in I$ and $b\in U_i\cap U_j$. For

$i\in I$, let $\varphi_i:\overset{-1}{p}(U_i)\rightarrow (^-{p'}^1)(U_i)$ be the mapping defined by $f'_i\circ \varphi_i\circ$ $f_i^{-1}(b, g) = (b, h_i(b)g)$, for $b\in U_i$ and $g\in G$. It is an isomorphism of principal fibre spaces with basis $U_i$ and group G. For $(i, j)\in$ $I\times I$, let us denote by $\gamma_{ij}$ and $\gamma '_{ij}$ the gluing homeomorphisms associated as above with the cocycles $(g_{ij})$ and $(g'_{ij})$ respectively (I, p. 115, formula (5)), so that $f_i(b, g) =\gamma_{ij}\circ f_j(b, g)$ and $f'_i(b, g) =\gamma '_{ij}\circ f'_j(b, g)$ for every $(b, g)\in (U_i\cap U_j)\times G$. Consequently, for $(i, j)\in I\times I$ and $(b, g)\in (U_i\cap U_j)\times G$, we have the relations

This proves that $\varphi_i$ and $\varphi_j$ coincide on $\overset{-1}{p}(U_i\cap U_j)$. The morphisms $\varphi_i$ therefore glue together to a B-morphism of principal fibre spaces from E to $E'$. Assertion b) follows, since every morphism of principal fibre spaces with basis B and group G is an isomorphism (I, p. 93, Prop. 1).

Let us now prove a). Let $(E, p)$ be a principal fibre space with group G endowed, for each $i\in I$, with a trivialization $f_i:\overset{-1}{p}(U_i)\rightarrow$ $U_i\times G$ over $U_i$. Let $(g_{ij})$ be the cocycle defined by this family, and let F then be the principal fibre space defined by the cocycle $(g_{ij})$. By construction, the principal fibre space F is endowed with a trivialization over $\mathscr{U}$; this trivialization defines the cocycle $(g_{ij})$. By assertion b), the principal fibre space $(E, p)$ is isomorphic to F.

Let B be a topological space and G a topological group. Let $(E, p)$ be a principal fibre space with basis B and group G. Let $s$ be a section of the surjective mapping $p$ (E, II, p. 19, Prop. 8). The mapping $f: B\times G\rightarrow E$ defined by $f(b, g) =s(b)\cdot g$ is a bijection compatible with the action of G. Endow $B\times G$ with the topology obtained by transport of structure; $(B\times G$, pr$_1)$ is then a principal fibre space with basis B and group G isomorphic to E. There therefore exists a set T of principal fibre spaces with basis B and group G such that every principal fibre space with basis B and group G is isomorphic to an element of T. Let us denote by $P(B,G)$ the set of isomorphism classes of principal fibre spaces with basis B and group G (E, II, p. 47).

Let $\mathscr{U}= (U_i)_{i\in I}$ be an open covering of B. Let $P(B,\mathscr{U},G)$ denote the subset of $P(B,G)$ consisting of the isomorphism classes of principal bundles which are trivializable over $\mathscr{U}$. By proposition 11, there exists a mapping $r_{\mathscr{U}}$ of $H^1_{cont}(B,\mathscr{U},G)$ into $P(B,\mathscr{U},G)$ which associates with the class of a cocycle $(g_{ij})\in Z^1_{cont}(B,\mathscr{U},G)$ the isomorphism class of the principal fibre space defined by this cocycle; it is a bijection (loc. cit.). The inverse bijection $s_{\mathscr{U}}$ associates with the isomorphism class in $H^1_{cont}(B,\mathscr{U},G)$ of a principal fibre space E, trivializable over $\mathscr{U}$, the class of the cocycle defined by an arbitrary family of trivializations of E over $\mathscr{U}$. Under this correspondence, the isomorphism class of the trivial principal fibre space $B\times G$ corresponds to the cohomology class of the constant cocycle $g_{ij}=e$, also called the trivial cocycle.

By virtue of the definition of a principal fibre space, the set $P(B,G)$ is the union of the sets of the form $P(B,\mathscr{U},G)$, where $\mathscr{U}$ is an open covering of B.

Let $\mathscr{U}= (U_i)_{i\in I}$ be an open covering of B and let $\mathscr{V}= (V_k)_{k\in K}$ be an open covering of B finer than $\mathscr{U}$. We have $P(B,\mathscr{U},G)\subset$ $P(B,\mathscr{V},G)$; let $i_{\mathscr{V} \mathscr{U}}$ denote the canonical injection defined by this inclusion. Choose a mapping $\varphi : K\rightarrow I$ such that $V_k\subset U_{\varphi(k)}$ for every $k\in K$. Given a cocycle $(g_{ij})\in Z^1_{cont}(B,\mathscr{U},G)$, set, for every pair $(k, \ell )\in K\times K,\overline{g}_{k\ell}=g_{\varphi(k)\varphi(\ell)}|V_k\cap V_{\ell}$.

The family $(\overline{g}_{k\ell})$ is a cocycle on B, with values in G, subordinate to $\mathscr{V}$. If $(g'_{ij})\in Z^1_{cont}(B,\mathscr{U},G)$ is a cocycle cohomologous to the cocycle $(g_{ij})$, the cocycle $(\overline{g}'_{k\ell})$ deduced from $(g'_{k\ell})$ is cohomologous to the cocycle $(\overline{g}_{k\ell})$. It follows that there is a mapping

which associates with the class of $(g_{ij})$ the class of $(\overline{g}_{k\ell})$.

Let E be a principal fibre space with base B and group G and, for every $i\in$ I, let $f_i: E_{U_i}\rightarrow U_i\times G$ be a trivialization of the principal $U_i$-bundle $E_{U_i}$. For every $k\in$ K, let $f'_k: E_{V_k}\rightarrow V_k\times G$ be the trivialization deduced from $f_{\varphi(k)}$ by passing to subsets. Let $(g_{ij})\in Z^1_{cont}(B,\mathscr{U},G)$ be the cocycle defined by the family $(f_i)$; the cocycle defined by the family $(f'_k)$ is precisely the cocycle $(\overline{g}_{k\ell})$ defined above. Thus, the following diagram is commutative:

$r_{\mathscr{U}}r_{\mathscr{V}}$

Since the mappings $r_{\mathscr{U}}$ and $r_{\mathscr{V}}$ are bijective, the mapping $c(\varphi )$, analogously to $i_{\mathscr{V} \mathscr{U}}$, is an injection and does not depend on the choice of $\varphi$. Henceforth we shall write $c_{\mathscr{V} \mathscr{U}}$ instead of $c(\varphi )$.

Let $\mathscr{R}$ be the set of elements of $\mathfrak{P}(\mathfrak{P}(B))$ which are open coverings of B. For every open covering $\mathscr{U}$ of B, there exists an open covering $\mathscr{V}$ belonging to $\mathscr{R}$ such that $\mathscr{U}$ is both finer and coarser than $\mathscr{U}$. The set $\mathscr{R}$ is ordered and filtering for the relation $\leqslant$ defined by $\mathscr{U}\leqslant \mathscr{V}$ if $\mathscr{V}$ is a covering finer than $\mathscr{U}$. It follows from the foregoing that an inductive system $(H^1_{cont}(B,\mathscr{U},G), c_{\mathscr{V} \mathscr{U}})$ relative to the filtered ordered set $\mathscr{R}$ has been defined and that the family $(r_{\mathscr{U}})$ is an inductive system of bijective mappings of $(H^1_{cont}(B,\mathscr{U},G), c_{\mathscr{V} \mathscr{U}})$ into $(P(B,\mathscr{U},G), i_{\mathscr{V} \mathscr{U}})$. If we denote by $H^1_{cont}(B,G)$ the inductive limit of the system $(H^1_{cont}(B,\mathscr{U},G), c_{\mathscr{V} \mathscr{U}})$ and by $r: H^1_{cont}(B,G)\rightarrow P(B,G)$ the inductive limit of the family $(r_{\mathscr{U}})$, we therefore have: