Calculating homology groups

It should be kept in mind that although homology, like most of algebraic topology, is geometrically inspired, its algebraic constructions may or may not have ready geometric interpretations in odd situations or higher dimensions. Despite the “sensible” results above, counter-intuitive facts caution us to always deduce these and other measurements of a space with mathematical rigor. For example, the “hole” interpretation of homology does not easily extend to non-orientable surfaces:

  • \({H_{1}\left(\mathbb{R}\textrm{P}^{2}\right)=\mathbb{Z}_{2}}\), since if we view \({\mathbb{R}\textrm{P}^{2}}\) as \({D^{2}}\) with boundary \({S^{1}}\) having antipodal points identified, the path connecting two antipodal points is a “loop,” but any two such loops together is homologous to a point
  • \({H_{2}\left(\mathbb{R}\textrm{P}^{2}\right)=0}\), since any triangulation of \({\mathbb{R}\textrm{P}^{2}}\) has a boundary, the real projective plane being non-orientable
  • In general, any non-orientable manifold \({M^{n}}\) has \({H_{n}(M^{n})=0}\)

As one might guess from the examples considered so far, it is also a fact that the topology of 2-manifolds is completely determined by homology. This circumstance is certainly not true in higher dimensions, as we noted in the introduction to this chapter.

There are various relations that can help in calculating homology groups. For example, an immediate result is that if \({X}\) has connected components \({X_{i}}\), \({H_{n}(X)=\bigoplus_{i}H_{n}\left(X_{i}\right)}\). Another is the excision theorem, which states that for \({Z\subset A\subset X}\), \({H_{n}\left(X-Z,A-Z\right)\cong H_{n}\left(X,A\right)}\). Here \({H_{n}\left(X,A\right)}\) is a relative homology group, defined using \({n}\)-chains \({C_{n}\left(X,A\right)\equiv C_{n}\left(X\right)/C_{n}(A)}\) in place of \({C_{n}\left(X\right)}\); this construction essentially ignores any holes in \({A\subset X}\). Thus the excision theorem states the intuitively expected fact that we can delete any portion of a subspace \({A}\) without affecting \({H_{n}\left(X,A\right)}\), which already ignores holes in \({A}\).

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