There are many homological tools used in algebraic topology. Some variants of the homology groups include:
- Homology group with coefficients: for an abelian group G, Hn(X;G) is defined using n-chains Cn(X;G) with coefficients in G instead of Z
- Reduced homology groups: a slight variant that avoids the result H0=Z for points while keeping the higher homology groups the same
- Local homology of X at A: Hn(X∣A)≡Hn(X,X−A) depends only on a neighborhood of A in X
- Simplicial homology, cellular homology, etc.: more easily constructed homology theories that are only valid for certain types of spaces; all can be shown to be equivalent to singular homology for those spaces
- Cohomology groups: Hn(X;G) are dual constructions based on the cochain groups Cn(X;G)≡C∗n=Hom(Cn,G), the group of homomorphisms from Cn to some abelian group G; a homomorphism Hn(X;G)→Hom(Hn(X;G),G) can be constructed which is surjective, becoming an isomorphism if G is a field
- Cohomology ring: H∗(X;R) is a direct sum of the cohomology groups Hn(X;R) with coefficients in a ring R; multiplication is defined using the cup product, a product between the Hn(X;R)
Some related constructions include:
- Betti number: bn≡ the number of Z summands if Hn(X) is written Z⊕⋯⊕Zc1⊕Zc2⊕Zc3⊕⋯, where the ci are called torsion coefficients
- Euler characteristic: the alternating sum of Betti numbers χ=b0−b1+b2−⋯; for a cell complex, the number of even cells minus the number of odd cells, so that a compact connected surface has genus g=(2−χ)/2
- Brouwer degree (AKA winding number for S1): any mapping ϕ:Sn→Sn induces a homomorphism on Hn(Sn)=Z of the form z→az; the integer a is the Brouwer degree of the map, essentially the number of times the mapping wraps around the sphere
- Moore space: given an abelian group G and an integer n>0, the space M(G,n) is constructed to have Hn=G and Hi=0 for i≠n