Theorem 1

Let G be a topological group, let $B_u$ be a topological space, and let $(E_u, p_u)$ be a principal fibre space with group G and base $B_u$. Suppose that the space $E_u$ is homotopic to a point.

Let B be a paracompact topological space.

a) Every principal fibre space with group G and base B is isomorphic to a principal fibre space of the form $f^*E_u$, where $f: B\rightarrow B_u$ is a continuous mapping.

b) Let $f_0$ and $f_1$ be continuous mappings of B into $B_u$. In order that $f_0^*E_u$ and $f_1^*E_u$ should be isomorphic principal fibre spaces with group G and base B, it is necessary and sufficient that the mappings $f_0$ and $f_1$ should be homotopic.

In other words, there exists a mapping of $[B,B_u]$ into P(B; G) which associates with the homotopy class of a continuous mapping $f$ of B into $B_u$ the isomorphism class of the principal fibre space $f^*E_u$. This mapping is bijective.

Let E be a principal fibre space with group G and base B. By Prop. 4, the sheaf $\mathscr{F}$ on B of morphisms of principal fibre spaces of E into $E_u$ is soft. Hence $\mathscr{F}(B)$ is not empty, whence a).

Let $f_0$ and $f_1$ be continuous mappings of B into $B_u$. If the mappings $f_0$ and $f_1$ are homotopic, the principal fibre spaces $f_0^*E_u$ and $f_1^*E_u$ are isomorphic (IV, p. 445, corollary 2). Let us prove the converse. For $\alpha \in \{0,1\}$, let $E_{\alpha}$ denote the principal fibre $B_u$-space $f_{\alpha}^*E_u$, $g_{\alpha}: E_{\alpha}\rightarrow E_u$ the first projection and $p_{\alpha}: E_{\alpha}\rightarrow B$ the second projection. Let $i: E_0\rightarrow E_1$ be an isomorphism of principal fibre spaces. Let $p$ be the mapping $p_0\times$ Id$_{\mathbf{I}}: E_0\times \mathbf{I}\rightarrow B\times \mathbf{I}$.

Since the space $B\times \mathbf{I}$ is paracompact (TG, IX, p. 70, proposition 17), the sheaf $\mathscr{G}$ on $B\times \mathbf{I}$ of morphisms of principal fibre spaces from $E_0\times \mathbf{I}$ into $E_u$ is soft (IV, p. 446, proposition 4). Put $A = B\times \{0,1\}$, $U = B\times ([0,\frac{1}{2}[\cup ]\frac{1}{2},1])$, and define an element $g$ of $\mathscr{G}(U)$ by setting

$g_0(x)$ for $(x, t)\in E_0\times [0,^1[$,

$g(x, t) =$ 2

$g_1\circ i(x)$ for $(x, t)\in E_0\times ]\frac{1}{2},1]$.

Since the sheaf $\mathscr{G}$ is soft, there exist an element $h\in \mathscr{G}(B\times \mathbf{I})$ and an open neighbourhood V of A in U such that $h|V =g|V$; such an element is a continuous mapping $H: E_0\times \mathbf{I}\rightarrow E_u$, compatible with the operations of G and such that $H(x,0) =g_0(x)$, $H(x,1) =g_1(i(x))$ for every $x\in E_0$. There then exists a mapping $h': B\times \mathbf{I}\rightarrow B_u$ such that $h'(p_0(x), t) =p_u(H(x, t))$ for $x\in E_0$ and $t\in \mathbf{I}$; this mapping is continuous, and it is a homotopy joining $f_0$ to $f_1$.