Definition 6

Let B be a topological space, let G be a topological group, and let $\mathscr{U}= (U_i)_{i\in I}$ be an open covering of B. A continuous 1-cocycle on B with values in G, subordinate to $\mathscr{U}$, is by definition the data, for each pair $(i, j)$ of elements of I, of a continuous mapping $g_{i,j}$ of $U_i\cap U_j$ into G, such that for every triplet $(i, j, k)\in I\times I\times I$ and every point $b$ of $U_i\cap U_j\cap U_k$, one has

We denote by $Z^1_{cont}(B,\mathscr{U},G)$ the set of continuous 1-cocycles on B with values in G, subordinate to the covering $\mathscr{U}$.

In this section, we shall simply say cocycle instead of continuous 1-cocycle.

Let $(E, p)$ be a principal fibre space over B of group G, and let $\mathscr{U}$ = $(U_i)_{i\in I}$ be a covering of B by open sets $U_i$. One says that E is trivializable over $\mathscr{U}$ if, for every $i\in I$, E is trivializable over $U_i$. One then calls a $\mathscr{U}$ -trivialization of E a family $(f_i)_{i\in I}$, where $f_i:\overset{-1}{p}(U_i)\rightarrow U_i\times G$ is a trivialization of E over $U_i$.

Let $\mathscr{U}= (U_i)_{i\in I}$ be a covering of B by open sets, and let $(E, p)$ be a principal fibre space over B of group G endowed with a $\mathscr{U}$ -trivialization $(f_i)_{i\in I}$. For every pair $(i, j)\in I\times I$, let us denote by $g_{ij}$ the mapping of $U_i\cap U_j$ into G defined by

where $e$ denotes the identity element of G. It is continuous.

Since $f_i$ and $f_j$ are compatible with the operations of G, one has

for $b\in U_i\cap U_j$ and $g\in G$. If $b$ is a point of $U_i\cap U_j\cap U_k$ (where $i,j$, $k\in I$), one has

and

It follows that

so that the family $(g_{ij})$ is a cocycle on B with values in G, subordinate to the covering $\mathscr{U}$. It is called the cocycle defined by the family of trivializations $(f_i)_{i\in I}$.

Let B, G and $\mathscr{U}$ be as in definition 6, and let $(g_{ij})\in$ $Z^1_{cont}(B,\mathscr{U},G)$ be a cocycle. For every pair $(i, j)\in I\times I$, the mapping $\gamma_{ij}: (U_i\cap U_j)\times G\rightarrow (U_i\cap U_j)\times G$ defined by

(5) $\gamma_{ij}(b, g) = (b, g_{ij}(b)g)$ for $b\in U_i\cap U_j$ and $g\in G$

is an isomorphism of principal fibre spaces with base $U_i\cap U_j$. For every triplet $(i, j, k)\in I\times I\times I$ and every pair $(b, g)\in (U_i\cap U_j\cap U_k)\times G$, one has :

Let F be the topological space obtained by gluing the spaces $U_i\times G$ along the $(U_i\cap U_j)\times G$ by means of the bijections $\gamma_{ij}$ (TG, I, p. 16). For each $i\in I$, the image of $U_i\times G$ in F is an open set of F (TG, I, p. 17, Prop. 9). The canonical projections $p_i: U_i\times G\rightarrow U_i$ glue together to give a continuous mapping $p$ of F into B. Since the mappings $\gamma_{ij}$ are compatible with the right operations of G in the spaces $U_i\times G$, one deduces a continuous law of operation of G on the right in F which makes F into a principal fibre space with base B and group G, endowed with a trivialization over each $U_i$. One says that F is the principal fibre space defined by the cocycle $(g_{ij})$.