Cellular homology
(Cellular homology)
For a CW complex \(X\), the singular homology \(H_*(X)\) is the homology of the cellular chain complex, whose group \(C_n\) is free abelian on the \(n\)-cells. The coefficient of an \((n-1)\)-cell \(e^{n-1}_\beta\) in \(\del e^n_\alpha\) is the degree of the composite of the attaching map \(S^{n-1}\to X^{(n-1)}\) with the quotient \(X^{(n-1)}\to X^{(n-1)}/\qty{X^{(n-1)}\sm e^{n-1}_\beta}\cong S^{n-1}\).
- If \(X\) has no cells in any two adjacent dimensions, then every cellular boundary map is zero, so \(H_n(X)\) is free on the \(n\)-cells; for \(\CP^n\) this gives \(\ZZ\) in each even degree \(0,2,\ldots,2n\).
- If \(X\) has no cells above dimension \(d\), then \(H_n(X) = 0\) for \(n>d\).
The Mayer–Vietoris sequence
(Mayer--Vietoris)
For subspaces \(A,B\subseteq X\) whose interiors cover \(X\), there is a long exact sequence \[ \cdots \to H_n(A\intersect B) \to H_n(A)\oplus H_n(B)\to H_n(X)\to H_{n-1}(A\intersect B)\to\cdots. \]
(Standard decompositions)
\(S^n\) as the union of two open hemispherical caps meeting in a band \(\homotopic S^{n-1}\); a connected sum \(M\# N\) of \(n\)-manifolds as the union of \(M\) and \(N\) each minus an open ball, meeting in \(S^{n-1}\); the torus as the union of two cylinders meeting in two disjoint circles.
Pairs and quotients
For a pair \((X,A)\) there is a long exact sequence \[ \cdots\to H_n(A)\to H_n(X)\to H_n(X,A)\to H_{n-1}(A)\to\cdots, \] and if \((X,A)\) is a good pair, the quotient map induces \(H_n(X, A)\cong \tilde H_n(X/A)\) for all \(n\).
The reduced homology of a quotient \(X/A\) of a good pair is therefore computed from the long exact sequence of \((X,A)\).
Choosing a method
| The space is given as | Method |
|---|---|
| a cell complex, or a polygon with identifications | cellular homology |
| a union of two subspaces with known homology | Mayer–Vietoris |
| a quotient \(X/A\) of a good pair, or a pair | the long exact sequence of the pair |
| a product | the Künneth theorem |
Consistency checks
- If \(X\) has finitely many path components, \(H_0(X) \cong \ZZ^{\abs{\pi_0(X)}}\).
- For path-connected \(X\), \(H_1(X)\) is the abelianization of \(\pi_1(X)\), which can be compared with a fundamental group computation.
- For a finite CW complex \(X\) with \(c_n\) cells of dimension \(n\), \(\chi(X)=\sum_n (-1)^n \operatorname{rank} H_n(X) = \sum_n (-1)^n c_n\).
- For a connected closed \(n\)-manifold \(M\): if \(M\) is orientable, \(H_n(M) \cong \ZZ\) and \(H_{n-1}(M)\) is free; if \(M\) is non-orientable, \(H_n(M) = 0\) and the torsion subgroup of \(H_{n-1}(M)\) is \(\ZZ/2\). The torus and the Klein bottle have \(H_1\cong\ZZ^2\) and \(H_1\cong\ZZ\oplus\ZZ/2\).