[at/2] Cellular Homology


Cellular homology

Cellular homology provide an efficient way to compute homology groups of CW complexes, based on degree calculations.

Let \(X\) be a CW complex; then:

  1. \(H_k(X^n, X^{n - 1})\) is zero for \(k \neq n\), and free abelian otherwise, with basis in one-to-one correspondence with the n-cells of \(X\). This follows from the observation that \((X^n, X^{n - 1})\) is a good pair, and \(X^n/X^{n - 1}\) is the wedge sum of \(n\)-spheres, one for each \(n\)-cell of \(X^n\).
  2. \(H_k(X^n) = 0\) for \(k \gt n\).
  3. The inclusion \(i : X^n \hookrightarrow X\) induces the isomorphism \(i_* : H_k(X^n) \rightarrow H_k(X)\) if \(k \lt n\), since \(H_k(X^n) \approx H_k(X^{n + 1}) \approx \ldots H_k(X^{n + m})\) for all \(m \geq 0\), as in (b).

To prove (b), consider the l.e.s of the pair \((X^n, X^{n - 1})\):

\[H_{k + 1}(X^n, X^{n - 1}) \rightarrow H_k(X^{n - 1}) \rightarrow H_k(X^n) \rightarrow H_k(X^n, X^{n - 1})\]

If \(k \neq n, n - 1\), then the outer two groups are zero, and we get \(H_k(X^{n - 1}) \approx H_k(X^n)\). If \(k > n\), \(H_k(X^n) \approx H_k(X^{n - 1}) \approx \ldots \approx H_k(X^0)\).

Pairs \((X^{n + 1}, X^n), (X^n, X^{n - 1}), (X^{n - 1}, X^{n - 2})\) fit into the l.e.s:

\[ \begin{xy} \xymatrix{ & & & 0 & \\ 0\ar[dr] & & H_n(X^{n + 1})\ar[ur] & & \\ & H_n(X^n)\ar[dr]^{j_n}\ar[ur] & & & \\ H_{n + 1}(X^{n + 1}, X^n)\ar[ur]^{\partial_{n + 1}}\ar[rr]^{d_{n + 1}} & & H_n(X^n, X^{n - 1})\ar[dr]^{\partial_n}\ar[rr]^{d_n} & & H_n(X^{n - 1}, X^{n - 2}) \\ & & & H_{n - 1}(X^{n - 1})\ar[ur]^{j_{n - 1}} & \\ & & 0\ar[ur] & & } \end{xy} \]

\(d_{n + 1}, d_n\) are relativizations of the boundary map \(\partial_{n + 1}, \partial_n\), and the composition \(d_n d_{n + 1}\) is zero. The horizontal row in the diagram is termed `cellular chain complex` in \(X\). Since \(H_n(X^n, X^{n - 1})\) is free with basis in one-to-one correspondence with the \(n\)-cells, it can be thought of as linear combinations of \(n\)-cells of \(X\). The homology groups of this cellular chain complex are called `cellular homology groups`.

\(H_n^\text{CW}(X) \approx H_n(X)\). From the diagram, \(H_n(X)\) can be identified with \(H_n(X^n)/\text{im } \partial_{n + 1}\). Further, since \(j_n\) is injective, it maps \(\text{im } \partial_{n + 1}\) isomorphically onto \(\text{im }(j_n \partial_{n + 1}) = \text{im } d_{n + 1}\), and \(H_n(X^n)\) onto \(\text{im } j_n = \text{ker } \partial_n\). Since \(j_{n - 1}\) is injective, \(\text{ker } d_n = \text{ker } \partial_n\). Thus, \(j_n\) induces a isomorphism of the quotient \(H_n(X^n)/\text{im } \partial_{n + 1}\) onto \(\text{ker } d_n/\text{im } d_{n + 1}\).

  1. \(H_n(X) = 0\) if \(X\) is a CW complex with no \(n\)-cells.
  2. If \(X\) is a CW complex with \(k\) \(n\)-cells, \(X\) is generated by atmost \(k\) elements; \(H(X^n, X^{n - 1})\) is free abelian on \(k\) generators, so \(\text{ker } d_n\) must be generated by at most \(k\) elements, hence also the quotient \(\text{ker } d_n/\text{im } d_{n + 1}\).
  3. If \(X\) is a CW complex with no two cells in adjacent dimensions, then it is free abelian with basis in one-to-one correspondence with the \(n\)-cells of \(X\). This is because the cellular boundary maps \(d_n\) are zero.

\(\mathbb{C}P^n\) has a CW-structure with each cell of each even dimension, where \(\mathbb{C}P^n\) is the complex projective \(n\)-space of complex lines through the origin in \(\mathbb{C}^{n + 1}\).

\[H_i(\mathbb{C}P^n) = \begin{cases} \mathbb{Z} \quad \text{for even i} \\ 0 \quad \text{otherwise} \end{cases}\]

As another example, \(S^n \times S^n\) has a product CW-structure, with one \(0\)-cell, two \(n\)-cells, and one \(2n\)-cell.

To compute cellular boundary maps \(d_n\), note that \(d_1 : H_1(X^1, X^0) \rightarrow H_0(X^0)\) is the same as the simplicial boundary map \(\Delta_1(X) \rightarrow \Delta_0(X)\). If \(X\) is connected, and has only one \(0\)-cell, \(d_1\) must be zero, otherwise \(H_0(X)\) would not be \(\mathbb{Z}\). When \(n \gt 1\), \(d_n\) can be computed in terms of degrees:

`Cellular boundary formula`: \(d_n(e_\alpha^n) = d_{\alpha \beta} e_\beta^{n - 1}\) where \(d_{\alpha \beta}\) is the degree of the map \(S_\alpha^{n - 1} \rightarrow X^{n - 1} \rightarrow S_\beta^{n - 1}\) that is the composition of attaching map \(e_\alpha^n\) with the quotient map collapsing \(X^{n - 1} - e_\beta^{n - 1}\) to a point. The cellular boundary formula can then be obtained from the following diagram:

\[ \begin{xy} \xymatrix{ H_n(D_\alpha^n, \partial D_\alpha^n)\ar[d]\ar[r] & \tilde{H}_{n - 1}(\partial D_\alpha^n)\ar[d]\ar[r] & \tilde{H}_{n - 1}(S_\beta^{n - 1})\ar[d] \\ H_n(X^n, X^{n - 1})\ar[dr]\ar[r] & \tilde{H}_{n - 1}(X^{n - 1})\ar[d]\ar[r] & \tilde{H}_{n - 1}(X^{n - 1}/X^{n - 2})\ar[d] \\ & H_{n - 1}(X^{n - 1}, X^{n - 2})\ar[r] & H_{n - 1}(X^{n - 1}/X^{n - 2}, X^{n - 2}/X^{n - 2}) } \end{xy} \]