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 \({G}\), \({H_{n}\left(X;G\right)}\) is defined using \({n}\)-chains \({C_{n}\left(X;G\right)}\) with coefficients in \({G}\) instead of \({\mathbb{Z}}\)
  • Reduced homology groups: a slight variant that avoids the result \({H_{0}=\mathbb{Z}}\) for points while keeping the higher homology groups the same
  • Local homology of \({X}\) at \({A}\): \({H_{n}\left(X\mid A\right)\equiv H_{n}\left(X,X-A\right)}\) 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: \({H^{n}\left(X;G\right)}\) are dual constructions based on the cochain groups \({C^{n}\left(X;G\right)\equiv C_{n}^{*}=\textrm{Hom}\left(C_{n},G\right)}\), the group of homomorphisms from \({C_{n}}\) to some abelian group \({G}\); a homomorphism \({H^{n}\left(X;G\right)\to\mathrm{Hom}\left(H_{n}\left(X;G\right),G\right)}\) can be constructed which is surjective, becoming an isomorphism if \({G}\) is a field
  • Cohomology ring: \({H^{*}\left(X;R\right)}\) is a direct sum of the cohomology groups \({H^{n}\left(X;R\right)}\) with coefficients in a ring \({R}\); multiplication is defined using the cup product, a product between the \({H^{n}\left(X;R\right)}\)

Some related constructions include:

  • Betti number: \({b^{n}\equiv}\) the number of \({\mathbb{Z}}\) summands if \({H_{n}(X)}\) is written \({\mathbb{Z}\oplus\dotsb\oplus\mathbb{Z}_{c_{1}}\oplus\mathbb{Z}_{c_{2}}\oplus\mathbb{Z}_{c_{3}}\oplus\dotsb}\), where the \({c_{i}}\) are called torsion coefficients; in a cell complex, \({b^{n}}\) is just the number of \({n}\)-cells
  • Euler characteristic: the alternating sum of Betti numbers \({\chi=b^{0}-b^{1}+b^{2}-\dotsb}\), i.e. for a cell complex the number of even cells minus the number of odd cells; thus a compact connected surface has genus \({g=\left(2-\chi\right)/2}\)
  • Brouwer degree (AKA winding number for \({S^{1}}\)): any mapping \({\phi\colon S^{n}\to S^{n}}\) induces a homomorphism on \({H_{n}(S^{n})=\mathbb{Z}}\) of the form \({z\rightarrow az}\); 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 \({M\left(G,n\right)}\) is constructed to have \({H_{n}=G}\) and \({H_{i}=0}\) for \({i\neq n}\)

An Illustrated Handbook