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Related constructions and facts

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(XA)Hn(X,XA) 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)Cn=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 ZZc1Zc2Zc3, where the ci are called torsion coefficients
  • Euler characteristic: the alternating sum of Betti numbers χ=b0b1+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=(2χ)/2
  • Brouwer degree (AKA winding number for S1): any mapping ϕ:SnSn induces a homomorphism on Hn(Sn)=Z of the form zaz; 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 in

An Illustrated Handbook