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:

   $$g_{ij}\cdot g_{jk}\cdot g_{ki}=e\quad\text{on }U_i\cap U_j\cap U_k.$$
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:

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