Singular Cohomology

Singular Cohomology

We can construct a specific cohomology theory, called singular cohomology, by dualizing the singular chain complex.

Cochains

Let be a topological space and be an abelian group (the coefficient group). The group of -cochains with coefficients in is the group of all homomorphisms from the singular -chain group to : An element is a map , and it is completely determined by its values on the basis of singular -simplices .

Coboundary Operator

The coboundary operator is defined as the dual of the boundary operator . For and a singular -simplex , it is defined by:

Lemma

. Therefore, is a cochain complex.

Proof For any and any -simplex : since in the singular chain complex.

Singular Cohomology Group

The -th singular cohomology group of with coefficients in is the -th cohomology group of this cochain complex: If the coefficient group , is clear from the context, or is not important for the discussion, we often write for .

A continuous map induces a chain map , which in turn induces a cochain map by . This map on cochains induces a homomorphism on the cohomology groups, .