Examples

The construction of the homology groups is somewhat complicated, but the idea behind it is quite intuitive. For simple spaces, we expect that \({H_{n}}\) will be a direct product of \({\mathbb{Z}}\) components, one for each “\({n}\)-dimensional hole” in \({X}\) that is not the boundary of a \({(n+1)}\)-volume, since each such hole can be wrapped around any number of times in either direction, and none of these wrappings are homologous.

28.homology-hole

Above, the cycle \({c_{1}}\) is the boundary of a disc, and so is homologous to a point. A hole in \({X}\) prevents \({c_{2}}\) from being the boundary of any surface. \({c_{3}}\) is homologous to \({c_{4}}\) since their difference is the boundary of an annulus, thus preventing the hole from being counted twice. Thus \({H_{1}\left(X\right)=\mathbb{Z}\oplus\mathbb{Z}}\). Note that a cycle around both holes (not depicted) would be homologous to the cycle \({c_{2}+c_{4}}\).

The best way to see this is to consider some examples.

Homology Group Diagram
\({H_{1}(S^{1})=\mathbb{Z}}\) homology-H1S1
A loop can be mapped to a circle by wrapping around any number of times in either direction
\({H_{1}(S^{2})=0}\)  homology-H1S2
Any circle on the sphere is the boundary of a disc on the sphere
\({H_{2}(S^{1})=0}\) homology-H2S1
Any sphere mapped along the edge of a circle is always the boundary of a ball also mapped along the circle
\({H_{1}(T^{2})=\mathbb{Z}\oplus\mathbb{Z}}\)  homology-H1T2
The two circles that make up the 1-skeleton of a torus are not homologous and thus a loop can wrap around either circle any number of times in either direction
\({H_{2}(T^{2})=\mathbb{Z}}\) homology-H2T2
A torus can be mapped to itself by wrapping around any number of times in either direction

These calculations reflect the close relationship between homology groups and cell complex structure. Since the \({n^{\textrm{th}}}\) homology group is defined in terms of \({n}\)-surfaces, we intuitively expect \({H_{n}(X)}\) to only depend on the \({(n+1)}\)-skeleton of \({X}\) (which recall includes all \({k}\)-cells for \({k\leq n+1}\)), and this is in fact true if \({X}\) is a cell complex. Thus if \({X}\) is a cell complex with finitely many cells, for example a closed manifold, \({H_{n}(X)}\) is a finitely generated group.

For example, inductively extending the first three observations to spheres in arbitrary dimension shows that \({H_{n}(S^{d})=\mathbb{Z}}\) if \({n=d}\), 0 otherwise. For the \({d}\)-dimensional torus \({T^{d}}\), \({H_{n}(T^{d})}\) is the direct sum of \({c}\) copies of \({\mathbb{Z}}\) with \({c}\) the binomial coefficient \({d}\) choose \({n}\).

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