Chapter 11 — Čech Cohomology Framework

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1. #1

   A principal -bundle is defined by an open cover with transition functions satisfying the Cech cocycle condition:

   $$g_{ij}\cdot g_{jk}\cdot g_{ki}=e\text{on }{U}_i\cap U_j\cap U_k.$$
2. #2

   For a cover and coefficient group :

   $$C^k(\mathcal{U};G):=\prod _{i_0<\cdots <i_k}G.$$
3. #3

   $$(\delta f{)}_{i_0\cdots i_{k+1}}:=\prod _{j=0}^{k+1}{f}_{i_0\cdots {\hat{i}}_j\cdots i_{k+1}}^{(-1{)}^j}.$$
4. #4

   Cohomology

   $${\check{H}}^k(\mathcal{U};G):=\ker {\delta }^k/\mathrm{im}{\delta }^{k-1}.$$
5. #5

   The full Cech cohomology is:

   $${\check{H}}^k(X;G):=\underset{\mathcal{U}}{\underrightarrow{\lim }}{\check{H}}^k(\mathcal{U};G).$$