Why functions $f_{\alpha \beta} \colon U_\alpha \cap U_\beta \to Y$ are called cocycles?

Question

Let $X$ be some smooth manifold and $\{U_\alpha\}$ be its open cover. The last month I hear very often that one calls a collection of functions $f_{\alpha \beta} \colon U_\alpha \cap U_\beta \to Y$, where $Y$ is equal to $\mathbb R$, $\mathbb Z$ (or even to some group, for example $GL_n(\mathbb R)$) "cocycles". For example, "Maslov cocycles", "Gelfand-Fuks cocycles", "glueing cocycles" (in the definition of vector bundles). But there is no explanation why is it called so. So any help and any reference is very appreciated. If $\{(U_\alpha,\varphi_\alpha)\}$ is the atlas for $X$ will the collection of transition functions $\varphi_\beta \circ \varphi_\alpha^{-1}$ be some cocycle in this sense?

P.S. I know about simplicial, singular, cell homologies and related cohomologies (i.e. groups of holomogies assosiated to cochain complex obtained by application of $\mathrm{Hom}(\cdot,G)$ functor to chain complex of appropriate chains: simplicial, singular, cell) but with no relation to manifolds.

Answer 1

This answer addresses the reason behind the term "gluing cocycle" for the transition functions $\theta_{\beta\alpha}: U_\alpha \cap U_\beta \longrightarrow G$ of a fiber bundle $E \longrightarrow X$ with structure group $G$ and fiber $F$. The short answer is that the set of transition functions correspond to a cocycle in what is known as the first Čech cohomology of $X$ (subordinate to the cover $\{U_\alpha\}$ of $X$ that we use to define the transition functions).

The three conditions

(i) $\theta_{\alpha\alpha}(x) = 1$ for all $x \in X$ and $\alpha \in A$.

(ii) $\theta_{\beta\alpha}(x) = \theta_{\alpha\beta}(x)^{-1}$ for all $x \in X$ and $\alpha, \beta \in A$.

(iii) $\theta_{\gamma\beta}(x)\theta_{\beta\alpha}(x) = \theta_{\gamma\alpha}(x)$ for all $x \in X$ and $\alpha, \beta, \gamma \in A$.

satisfied by all transition functions are related to the Čech cohomology of $X$ with coefficients in the constant sheaf $\underline{G}$ in the following way.

A $G$-valued Čech $1$-cocycle subordinate to the open cover $\{U_\alpha\}_{\alpha \in A}$ of $X$ is a collection of continuous maps $\{f_{\beta\alpha}\}_{\alpha, \beta \in A}$ satisfying conditions (i)-(iii) above. Two Čech cocycles $\{f_{\beta\alpha}\}_{\alpha, \beta \in A}$, $\{f_{\beta\alpha}^\prime\}_{\alpha, \beta \in A}$ are said to be cohomologous if there exists a family of continuous maps $\{g_\alpha\}_{\alpha \in A}$, $$g: U_\alpha \longrightarrow G,$$ such that $$g_\beta(x) f_{\beta\alpha}(x) = f_{\beta\alpha}^\prime(x) g_\alpha(x)$$ for all $x \in U_\alpha \cap U_\beta$ and $\alpha, \beta \in A$.

If $\{f_{\beta\alpha}\}_{\alpha, \beta \in A}$, $\{f_{\beta\alpha}^\prime\}_{\alpha, \beta \in A}$ are cohomologous, write $$\{f_{\beta\alpha}\}_{\alpha, \beta \in A} \sim \{f_{\beta\alpha}^\prime\}_{\alpha, \beta \in A}.$$ We define the first Čech cohomology of $X$ with coefficients in $G$ subordinate to the cover $\{U_\alpha\}_{\alpha \in A}$ by $$\check{H}^1(\{U_\alpha\}; G) = \text{Čech cocycles}/\!\sim.$$ The first Čech cohomology of $X$ with coefficients in $G$ is then the direct limit of the $\check{H}^1(\{U_\alpha\}; G)$ over all open covers: $$\check{H}^1(X; G) = \varinjlim \check{H}^1(\{U_\alpha\}; G).$$

Note that if $\pi: E \longrightarrow X$ and $\pi': E' \longrightarrow X$ are two $G$-bundles over $X$ with fiber $F$ and respective transition functions $\{\theta_{\beta\alpha}\}_{\alpha, \beta \in A}$ and $\{\theta_{\beta\alpha}^\prime\}_{\alpha, \beta \in A}$ (we implicitly assume both $E$ and $E'$ can be trivialized over $\{U_\alpha\}$), a bundle isomorphism $\varphi: E \longrightarrow E'$ locally determines maps $$\varphi_\alpha: U_\alpha \longrightarrow G$$ satisfying $$\varphi_\beta(x)\theta_{\beta\alpha}(x) = \theta_{\beta\alpha}^\prime(x) \varphi_\alpha(x)$$ for all $x \in U_\alpha \cap U_\beta$ and $\alpha, \beta \in A$. Explicitly, if $$\psi_\alpha: \pi^{-1}(U_\alpha) \longrightarrow U_\alpha \times F$$ is a trivialization of $E$ over $U_\alpha$ and $\psi_\alpha^\prime$ is a trivialization of $E'$ over $U_\alpha$, then $$\varphi_\alpha = \psi_\alpha^\prime \circ \varphi \circ \psi_\alpha^{-1}.$$ In other words, isomorphic bundles have cohomologous gluing cocycles.

This indicates the following.

Theorem. The set of isomorphism classes of fiber bundles over $X$ with fiber $F$ and structure group $G$ are in bijective correspondence with the first Čech cohomology $\check{H}^1(X; G)$.