Formalism of Homology and Cohomology

Homology with Coefficients

Chain Complex with Coefficients

Let be a topological space and be an abelian group, which we call the coefficient group. The singular chain group of with coefficients in is defined as the free -module with basis as the set of singular -simplices in . An element of is a finite formal sum of the form , where and are singular -simplices. The usual boundary map extends -linearly to a boundary map by defining: Since still holds, forms a chain complex.

Homology with Coefficients

The -th homology group of with coefficients in , denoted , is the -th homology group of the chain complex :

e.g.

• The “usual” singular homology is simply homology with integer coefficients. .
• The reduced homology of the -sphere with coefficients in is:
• Using the cellular chain complex for with coefficients in , we find:
• Popular choices for the coefficient group include:
  • for a prime .
  • The field of rational numbers .
  • The -adic integers .

Naturality and Splitting

Proposition

The connecting homomorphism in the long exact sequence of a pair is a natural transformation between the functors and from the category of pairs of spaces to the category of abelian groups .

Splitting via Retraction

If is a pair and there exists a retraction (i.e., a map such that where is the inclusion), then the long exact sequence of the pair splits. The map is injective, and we have a split short exact sequence for each : This implies that the homology of decomposes into a direct sum:

Axiomatic Approach

The properties of singular homology can be distilled into a set of axioms, known as the Eilenberg-Steenrod axioms.

Homology Theory

A homology theory is a sequence of functors for , equipped with natural transformations , satisfying the following axioms:

1. Homotopy Invariance: If , then .
2. Excision: For any such that , the inclusion induces an isomorphism .
3. Long Exact Sequence: For any pair , there is a long exact sequence:
4. Additivity: If is a disjoint union , then the inclusion maps induce an isomorphism .

If, in addition, the theory satisfies the Dimension Axiom: for some abelian group , then it is called an ordinary homology theory.

Remark

abbreviates .

Uniqueness of Homology

If is an ordinary homology theory with , then there is a natural isomorphism for any CW pair .

Proof Sketch: The proof relies on showing that the axioms uniquely determine the homology of CW complexes. One shows that singular homology satisfies the axioms. Then, for any other theory satisfying the axioms, one builds a natural isomorphism to inductively over the skeletal filtration of a CW complex, using the Five Lemma at each step.

Cohomology

The central idea of cohomology is to “dualize” the constructions of homology by applying a functor, which flips the direction of most of the maps involved. To make this precise, we first recall the notion of an opposite category:

The opposite or dual category of a category has the same objects as , and an arrow in is an arrow in . That is is just with all of the arrows formally turned around. It is convenient to have a notation to distinguish an object in from the same one in . Thus we may write in for in . With this notation we can define composition and units in in terms of the corresponding operations in , namely

Link to original

Cohomology Theory

A cohomology theory consists of a sequence of functors for each , and a sequence of natural transformations called the connecting homomorphisms, satisfying the following axioms:

1. Homotopy Invariance: If , then the induced maps are equal, .
2. Excision: The inclusion induces an isomorphism in cohomology.
3. Additivity: For a disjoint union of spaces, .
4. Long Exact Sequence: For any pair , there is a long exact sequence in cohomology:

Why study cohomology?

• It is another algebraic invariant of topological spaces, which sometimes contains different information than homology.
• The direct sum of cohomology groups, , can be given the structure of a graded ring via the cup product: . A continuous map induces a ring homomorphism .
• Poincaré Duality, a deep result for manifolds, relates the homology of a space to its cohomology.