Cohomology Groups

Cohomology is the algebraic dual of homology, obtained by applying the $\operatorname{Hom}$ functor to the chain complex. It carries additional multiplicative structure that makes it strictly more powerful than homology for many topological problems.

Cochain Complexes

Definition

Let $C_*(X)$ be the singular chain complex of a space $X$ and $G$ an abelian group. The singular cochain group with coefficients in $G$ is $C^n(X; G) = \operatorname{Hom}(C_n(X), G)$ The coboundary map $\delta^n : C^n(X; G) \to C^{n+1}(X; G)$ is defined by $(\delta^n \varphi)(\sigma) = \varphi(\partial_{n+1} \sigma)$ for any $(n+1)$ -singular simplex $\sigma$ and cochain $\varphi \in C^n(X; G)$ .

One verifies $\delta^{n+1} \circ \delta^n = 0$ since $\partial_n \circ \partial_{n+1} = 0$ , so $(C^*(X; G), \delta)$ forms a cochain complex.

Definition

The $n$ -th cohomology group of $X$ with coefficients in $G$ is $H^n(X; G) = \ker \delta^n / \operatorname{im} \delta^{n-1} = Z^n(X; G) / B^n(X; G)$ Elements of $Z^n = \ker \delta^n$ are called cocycles and elements of $B^n = \operatorname{im}\, \delta^{n-1}$ are called coboundaries.

Basic Properties

ExampleZeroth cohomology

For a space $X$ with path components $\{X_\alpha\}_{\alpha \in A}$ , we have $H^0(X; G) \cong \prod_{\alpha \in A} G$ . This is the group of locally constant $G$ -valued functions on $X$ . Note the product (not direct sum) — this is a key difference from $H_0(X; G) \cong \bigoplus_\alpha G$ .

RemarkContravariant functoriality

Cohomology is a contravariant functor: a continuous map $f : X \to Y$ induces $f^* : H^n(Y; G) \to H^n(X; G)$ in the reverse direction. This arises because $\operatorname{Hom}(-, G)$ is contravariant. The contravariance is essential for defining the cup product and ring structure on cohomology.

Cohomology satisfies the same Eilenberg-Steenrod axioms as homology (with arrows reversed), including homotopy invariance, excision, and the long exact sequence of a pair. These properties make it an equally valid but complementary algebraic invariant.