Reduced cohomology - Knowlpedia

Let $X$ be a nonempty topological space, $A$ an abelian group, and $H^k(X;A)$ its singular cohomology. For $k\geq0$ , the reduced cohomology group is

\widetilde H^k(X;A)= \begin{cases} H^0(X;A)/A,&k=0,\\ H^k(X;A),&k>0, \end{cases}

where the quotient is by the subgroup of constant $0$ -cocycles.

Augmented cochain description

Equivalently, dualize the augmented singular chain complex with coefficients in $A$ and take cohomology. The added coboundary sends $a\in A$ to the constant $0$ -cochain with value $a$ . This gives the natural reduction map $H^k(X;A)\to\widetilde H^k(X;A)$ , which is the quotient in degree zero and the identity in positive degrees.

For any chosen point $x_0\in X$ , the long exact sequence of the pair identifies these groups with relative cohomology $H^k(X,\{x_0\};A)$ . No path-connectedness assumption is needed. The quotient definition itself requires no choice of basepoint.

Basic examples

For a one-point space, $\widetilde H^k(\{x_0\};A)=0$ for all $k\geq0$ . For $n>0$ ,

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

Degree zero

Ordinary $H^0(X;A)$ is the group of functions from the set of path components of $X$ to $A$ . Reduced cohomology quotients this product by the diagonal subgroup of constant functions. In particular, it vanishes for a nonempty path-connected space.

Reference

Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002, §3.1, pp. 199–200, “Reduced Groups” and “Relative Groups.” Author-hosted chapter.