Singular cohomology group - Imported Knowlpedia Content

Definition

Let $X$ be a topological space and $A$ an abelian group. The singular $k$ -chains $C_k(X;\mathbb Z)$ form the free abelian group on continuous maps $\Delta^k\to X$ , with the alternating face boundary $\partial$ . The singular cochains with coefficients in $A$ are

C^k(X;A) := \operatorname{Hom}\!\bigl(C_k(X;\mathbb Z),A\bigr),

with coboundary $\delta\varphi=\varphi\circ\partial$ . The $k$ th singular cohomology group is

H^k(X;A) := \ker(\delta:C^k\to C^{k+1}) / \operatorname{im}(\delta:C^{k-1}\to C^k).

Thus it is the cohomology of the resulting cochain complex.

Functoriality and invariance

A continuous map $f:X\to Y$ induces a pullback

f^*:H^k(Y;A)\to H^k(X;A),

so singular cohomology is contravariant in the space. Homotopic maps induce the same pullback. Consequently, homotopy-equivalent spaces have isomorphic singular cohomology groups.

For a fixed $X$ , a homomorphism of coefficient groups $A\to B$ induces $H^k(X;A)\to H^k(X;B)$ . When $A=R$ is a commutative ring, cup products make

H^*(X;R)=\bigoplus_{k\ge0}H^k(X;R)

into a graded-commutative ring.

Examples

For a one-point space and any coefficient group $A$ , $H^0(\mathrm{pt};A)\cong A$ and $H^k(\mathrm{pt};A)=0$ for $k>0$ .
Every nonempty contractible space has the same cohomology as a point.
With integer coefficients,

H^k(S^n;\mathbb Z) \cong \begin{cases} \mathbb Z,&k=0,n,\\ 0,&\text{otherwise}, \end{cases}

for $n>0$ .

Coefficients and variants

The notation $H^k(X)$ often means $H^k(X;\mathbb Z)$ , but coefficients should be stated when ambiguity matters. Integral, rational, real, and finite-field coefficients can reveal different information. Reduced cohomology modifies degree zero so that a point has zero cohomology in every degree. Relative cohomology $H^k(X,B;A)$ records the topology of a pair $B\subseteq X$ and participates in a long exact sequence.

For a smooth manifold, integration of differential forms gives the de Rham comparison isomorphism

H^k_{\mathrm{dR}}(M)\cong H^k(M;\mathbb R).

This identifies real singular cohomology with de Rham cohomology, but not with integral cohomology: torsion classes disappear after passing to real coefficients.

References

Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002. Author-hosted chapter record. Relevant: Chapter 3, singular cohomology and cup products.
Edwin H. Spanier, Algebraic Topology, Springer, 1966. DOI record. Relevant: Chapter 5, cohomology theory.