Cohomology I

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

Detour on Homology

Recall that a homology group comes with two process:

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

CnCn1\cdots\rightarrow C_n \overset{\partial}{\rightarrow} C_{n-1} \overset{\partial}{\rightarrow}\cdots

  1. Determine the boundary BnZnCnB_n\subset Z_n\subset C_n and thus define the homology group to be


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

σ:ΔqX\sigma:\Delta^q\rightarrow X

where if XX 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 XX and its quotient X/AX/A

Hn(A)Hn(X)Hn(X,A)Hn1(A)\cdots\rightarrow H_n(A)\rightarrow H_n(X)\rightarrow H_n(X,A)\overset{\partial}{\rightarrow} H_{n-1}(A)\rightarrow\cdots

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

Hn(AB)ΦHn(A)Hn(B)ΨHn(X)Hn1(AB)\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

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

Hn(XZ,AZ)Hn(X,A),nN.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.

The Rise of Cohomology

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


to obtain a cochain complex, with a coboundary map

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

Then the cohomology group with coefficients in GG is defined by


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