Simplicial Homology

Our goal now is to define the simplicial homology groups of a \(\Delta\)-complex \(X\). Let \(\Delta_n(X)\) be the free abelian group with basis the open \(n\)-simplices \(e^n_\alpha\) of \(X\). Elements of \(\Delta_n(X)\), called \(\mathbf{n}\)-chains, can be written as finite formal sums \(\sum_\alpha n_\alpha e^n_\alpha\) with coefficients \(n_\alpha \in \mathbb{Z}\). Equivalently, we could write \(\sum_\alpha n_\alpha \sigma_\alpha\) where \(\sigma_\alpha: \Delta^n \rightarrow X\) is the characteristic map of \(e^n_\alpha\), with image the closure of \(e^n_\alpha\) as described above. Such a sum \(\sum_\alpha n_\alpha \sigma_\alpha\) can be thought of as finite collection, or ‘chain’, of \(n\)-simplices in \(X\) with integer multiplicities, the coefficients \(n_\alpha\).

As one can see in the next figure, the boundary of the \(n\)-simplex \([v_0,\cdots ,v_n]\) consists of the various \((n-1)\)-dimensional simplices \([v_0,\cdots,\hat{v}_i,\cdots,v_n]\), where the ‘hat’ symbol \(\hat\) over \(v-i\) indicates that this vertex is deleted from the sequence \(v_0,\cdots,v_n\). In terms of chains, we might then wish to say that the boundary of \([v_0,\cdots,v_n]\) is the \((n-1)\)-chain formed by the sum of the faces \([v_0,\cdots,\hat{v}_i,\cdots,v_n]\). However, it turns out to be better to insert certain signs and instead let the boundary of \([v_0,\cdots,v_n]\) be \(\sum_i (-1)^i[v_0,\cdots,\hat{v}_i,\cdots,v_n]\). Heuristically, the signs are inserted to take orientations into account, so that all the faces of a simplex are coherently oriented, as indicated in the following figure:

In the last case, the orientations of the two hidden faces are also counterclockwise when viewed from outside the \(3\)-simplex.

With this geometry in mind we define for a general \(\Delta\)-complex \(X\) a boundary homomorphism \(\partial_n : \Delta_n(X) \rightarrow \Delta_{n-1}(X)\) by specifying its values on basis elements:

\[\partial_n(\sigma_\alpha)=\sum_i(-1)^i\sigma_\alpha\mid[v_0,\cdots,\hat{v}_i,\cdots,v_n]\]

Note that the right side of this equation does indeed lie in \(\Delta_{n-1}(X)\) since each restriction \(\sigma_\alpha \mid [v_0,\cdots,\hat{v}_i,\cdots,v_n]\) is the characteristic map of an \((n-1)\)-simplex of \(X\).

Lemma 2.1. The composition \(\Delta_n(X) \xrightarrow{\partial_n}\Delta_{n-1}\xrightarrow{\partial_{n-1}}\Delta_{n-2}(X)\) is zero.

Proof: We have \(\partial_n(\sigma)=\sum_i(-1)^i\sigma\mid[v_0,\cdots,\hat{v}_i,\cdots,v_n]\), and hence

\[\begin{split}\begin{aligned} \partial_{n-1}\partial_n(\sigma)&=\sum_{j<i}(-1)^i(-1)^j\sigma\mid[v_0,\cdots,\hat{v}_j,\cdots,\hat{v}_i,\cdots,v_n]\\ \sum_{j>i}(-1)^i(-1)^{j-1}\sigma\mid[v_0,\cdots,\hat{v}_i,\cdots,\hat{v}_j,\cdots,v_n]\\\end{split}\]

The latter two summations cancel since after switching \(i\) and \(j\) in the second sum, it becomes the negative of the first. ◻

The algebraic situation we have now is a sequence of homomorphisms of abelian groups

\[\cdots \rightarrow C_{n+1} \xrightarrow{\partial_{n+1}} C_{n} \xrightarrow{\partial_{n+1}} C_{n-1} \rightarrow \cdots \rightarrow C_{1} \xrightarrow{\partial_{1}} C_{0} \xrightarrow{\partial_{0}} 0\]

with \(\partial_n \partial_{n+1}=0\) for each \(n\). Such a sequelnce is called a chain complex. Note that we have extended the sequence by a \(0\) at the right end, with \(\partial_0=0\). The equation \(\partial_n\partial_{n+1}=0\) is equivalent to the inclusion \(\text{Im}\partial_{n+1} \subset \text{Ker}\partial_n\), where Im and Ker denote image and kernel. So we can define the \(n^{th}\) homology group of the chain complex to be the quotient group \(H_n=\text{Ker}\partial_n/\text{Im}\partial_{n+1}\). Elements of \(\text{Ker}\partial_n\) are called cycles and elements of \(\text{Im}\partial_{n+1}\) are called boundaries. Elements of \(H_n\) are cosets of \(\text{Im}\partial_{n+1}\), called homology classes. Two cycles representing the same homology class are said to be homologous. This means their difference is a boundary.

Returning to the case that \(C_n=\Delta_n(X)\), the homology group \(\text{Ker}\partial_n /\text{Im}\partial_{n+1}\) will be denoted \(H^\Delta_n(X)\) and called the \(n^{th}\) simplicial homology group of \(X\).