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Let $X$ be a smooth compact $k$-surface in $\mathbb R^n$ without boundary. Today on my lection lecturer introduced Čech cohomology as follows (not like in Wikipedia): let $\mathcal U$ be a finite open cover of $X$ with sufficiently small balls. For each pair $U,V \in \mathcal U$, $U \cap V \neq \varnothing$ define a real number $C_{UV}$ such that the following properties hold: $$ C_{UV} + C_{VU} = 0, \\ C_{UV}+C_{VW}+C_{WU}=0, \;\;\; \text{if } U \cap V \cap W \neq \varnothing. $$ The collection $\{C_{UV}\}$ is called Čech 1-cocycle. For each $U \in \mathcal U$ define also a real number $\sigma_U$. Then this collection $\{\sigma_U\}$ is called Čech 0-cocycle. The 1-cocycle $\{C_{UV}\}$ is called coboundary if there is a 0-cocycle $\{\sigma_U\}$ such that for each $U,V \in \mathcal U$, $U \cap V \neq \varnothing$ we have $C_{UV} = \sigma_U - \sigma_V$. In this definition cocycles form a real vector space $\mathcal P$ and coboundaries form its subspace $\mathcal P_0$. Factorspace $\mathcal P / \mathcal P_0$ is called Čech 1-cohomology space. Then we defined action of Čech 1-cocycle on any smooth curve $\gamma$ on $X$ as a sum of all $C_{UV}$ for $U \cap V \cap \gamma \neq \varnothing$ and we have also showed that Čech 1-cohomology is isomorphic to de Rham 1-cohomology.

My question is why this definition differs so much from one given in Wikipedia? Definition given in Wikipedia looks very difficult, it takes a lot of steps and a lot of constructions (even inductive limit, but I think that this corresponds to requirement of smallness of cover $\mathcal U$ in my definition). Is there some book in which Čech cohomology is introduced like in my definition? The definition given by my lecturer seems to me very intuitive and I can't say this about definition given in Wikipedia. What is the relation of these two definitions for the case of smooth compact $k$-surfaces in $\mathbb R^n$? Do they agree?

Appliqué
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  • Wikipedia's definition defines Čech $n$-cohomology for all $n$, so it's more general. It also introduces concepts that are independently useful elsewhere in algebraic topology and mathematics. – Qiaochu Yuan Apr 24 '13 at 20:44
  • @QiaochuYuan Yes, I noted that it's more general (it uses presheaves e.t.c.). But we can introduce Čech $n$-cohomology for all $n$ using definition from my post (defining numbers $C_{U_1\ldots U_n}$ for each $U_1 \cap \ldots \cap U_n \neq \varnothing$ with appropriate cocycle conditions). Can I use my definition If I have a deal only with compact smooth surfaces in $\mathbb R^n$ (or maybe to get used to this concept of Cech colomology) or should I try to understand now the general definition? – Appliqué Apr 24 '13 at 20:51

1 Answers1

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Your lecturer defined the Čech cohomology of the cover $\mathcal U$ of $X$. Since the cover uses small enough balls it is a good cover, and gives the same result as any other good cover. The Wikipedia article begins the same way, at least now. I do not know what it said when you posted this question. And since good covers are cofinal in all covers of $X$ this cohomology will agree with the colimit definition. Then Wikipedia uses this to give the colimit definition which does not favor any one cover. I think a lot of texts do this but I have none at hand just now.

The colimit definition has the advantage, for theoretical purposes, of not needing to know which are the good covers and allowing manipulations that need not preserve good covers. (For specific calculations in a concrete case you will likely want to find a good cover.) And it generalizes to other contexts in ways that Qiaochu hints at -- for example Čech cohomology of etale sheaves.

You the OP may know all this by now, two years after posting, but it is a good question so I wanted to answer it.

Colin McLarty
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