Cohomology I

In this part we will go through the main idea of cohomology in algebraic topology.

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Detour on Homology

Recall that a homology group comes with two process:

1. Define the chain complex $\{C_*,\partial_*\}$, with a boundary map such that $\partial^2=0$.

$$\cdots\rightarrow C_n \overset{\partial}{\rightarrow} C_{n-1} \overset{\partial}{\rightarrow}\cdots$$

2. Determine the boundary $B_n\subset Z_n\subset C_n$ and thus define the homology group to be

$$H_n(C):=Z_n/B_n$$

After a somehow abstract definition, we introduce the notion of homology to CW-complex, from “simplicial chain” of a topological space, to the “singular chain” of a topological space defined by free Abelian group generated by continuous maps

$$\sigma:\Delta^q\rightarrow X$$

where if $X$ is a CW-complex, then the two homology are equivalent. Unfortunately (at least for Relue), the definition above cannot be used to compute the homology group of any concrete spaces, even after we have realized the fact that “Homology only depends on the homotopy type of the space” .

Thus, we ought to extract some theorems/tools/techniques from the definition to do “computation”. Some important theorems must be listed here.

1. Exact sequence: Reveal the basic relation between $X$ and its quotient $X/A$

$$\cdots\rightarrow H_n(A)\rightarrow H_n(X)\rightarrow H_n(X,A)\overset{\partial}{\rightarrow} H_{n-1}(A)\rightarrow\cdots$$

2. Mayer-Vietoris theorem: Acts as a “van-Kampen theorem” in homotopy theory

$$\cdots\rightarrow H_n(A\cap B)\overset{\Phi}{\rightarrow}H_n(A)\oplus H_n(B)\overset{\Psi}{\rightarrow} H_n(X)\overset{\partial}{\rightarrow} H_{n-1}(A\cap B)\rightarrow \cdots$$

3. Excision: Elementary characteristic of exact sequence. If $Z\subset A\subset X$ such that $\bar{Z}\subset A^\circ$, then the inclusion $i:(X-Z,A-Z)\rightarrow (X,A)$ induces an isomorphism

$$H_n(X-Z,A-Z)\approx H_n(X,A),\qquad \forall n\in\N.$$

The above are computations on homology. In cohomology theory, we also wish to derive theories like that – from definitions to techniques that are simple for computation.

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The Rise of Cohomology

Given a chain complex $\{C_*,\partial\}$ of free Abelian groups. Define its dualization

$$C^n:=\mathrm{Hom}(C_n,G).$$

to obtain a cochain complex, with a coboundary map

$$\begin{aligned} \delta:C^n&\rightarrow C^{n+1}\\ \alpha&\mapsto \alpha\circ\partial \end{aligned}$$

Then the cohomology group with coefficients in $G$ is defined by

$$H^n(C;G):=Z^n(C;G)/B^n(C;G).$$

[Fact] $\mathrm{Hom}(\cdot,G)$ is right exact.