Singular Homology

Singular Homology

Recall that a -complex is built out of simplices, and . Singular homology generalizes this by considering all continuous maps from the standard simplex into the space, not just those that are part of a specific simplicial decomposition.

Singular Chain Complex

For a topological space , the singular chain complex is a sequence of abelian groups and homomorphisms , called boundary maps:

• Singular -chains is the free abelian group generated by all continuous maps , where is the standard n-simplex.
• The boundary map is defined on a generator by: where is the restriction of to the -th face of .

A key property is that .

Singular Homology

An element of is called an -cycle, and an element of is called an -boundary. The -th singular homology group of is the quotient group:

Chain Map

A chain map between two chain complexes and is a sequence of homomorphisms such that the following diagram commutes:

Proposition

A continuous map induces a chain map by composition , which in turn induces a homomorphism on homology groups by .

Proof For each , and a singular -simplex , we have

Homotopy Invariance

It turns out that if two topological spaces are homotopic, then their homologies are isomorphic. To show this, we need the notion of a chain homotopy:

Chain Homotopy

Let be two chain maps. A chain homotopy between and is a sequence of homomorphisms such that for all :

Proposition

A homotopy from to induces a chain homotopy from to .

Proof Sketch Given a singular -simplex , we can define a map by . The “prism” can be subdivided into simplices of dimension . Let the vertices of be and the vertices of be . The prism is the union of -simplices for . The chain homotopy map is defined on a generator as: A direct (but lengthy) calculation shows that this definition of satisfies the chain homotopy equation . The calculation involves carefully tracking how the boundary of the prism relates to the faces of the individual -simplices in the subdivision. The “top” face corresponds to , the “bottom” face corresponds to , and the “side” faces correspond to .

Homotopy Invariance

If two maps are homotopic, then they induce the same homomorphism on homology groups:

Proof Suppose and are homotopic, and is the homotopy. By the previous proposition, the homotopy induces a chain homotopy from to so that For any homology class represented by a cycle , we have: Therefore, , which means , we conclude that .

Corollary

If and are homotopy equivalent, then their homology groups are isomorphic:

Homology of a Point

A simple but important calculation is the homology of a single point space, denoted .

Homology of a Point

The homology groups of a point are:

Proof For any , there is a unique continuous map . Thus, , generated by . The boundary map sends to: The resulting chain complex is: The homology of this complex is straightforward to compute. For , the kernel of one map is the image of the previous one, so . For , and , so .