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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.

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  • $\begingroup$ @HenryT.Horton Thank you, that clarifies a little the situation. But I think that in more than half of cases there were no talk about bundles when one speaked about cocycles. Then "cocycle conditions" are sufficient to form a Čech 2-cocycle for any collection of functions $f_{\alpha \beta} \colon U_\alpha \cap U_\beta \to Y$? And an arbitrary collection of functions $f_{\alpha \beta} \colon U_\alpha \cap U_\beta \to Y$ is a Čech 2-cochain? $\endgroup$ – Appliqué Apr 18 '13 at 19:55
  • $\begingroup$ There is something called Gelfand-Fuks cohomology that I know nothing about, but it is likely that the notion of a Gelfand-Fuks cocycle is related to that cohomology. I have not heard of a Maslov cocycle before, but there is something called a Maslov cycle which is not related to this question. Unless you have some references, I can only comment on gluing cocycles for the moment. $\endgroup$ – Henry T. Horton Apr 18 '13 at 20:06
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    $\begingroup$ I think the word cocycle arose in the context of cohomology, but it is also used outside of it. For example, the "compatibility" condition in the glueing lemma for schemes (diagram 2. here) is sometimes called the "cocycle condition". $\endgroup$ – A.P. Apr 18 '13 at 20:22
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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 $\check{H}^1(X; G)$.

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  • $\begingroup$ Very good answer! Thank you for clarification! $\endgroup$ – Appliqué Apr 18 '13 at 21:04

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