Complexes and their homology

 Preliminary exam prep

Here I'll present complexes from the most restrictive to the most general. Recall the standard $n$-simplex is
$$\Delta^n = \{x\in \R^{n+1}\ :\ \textstyle\sum x_i = 1, x_i\>0\}.$$

Definition: Let $V$ be a finite set. A simplicial complex $X$ on $V$ is a set of distinct subsets of $V$ such that if $\sigma\in X$, then all the subsets of $\sigma$ are in $X$.

Every $n$-simplex in a simplicial complex is uniquely determined by its vertices, hence no pair of lower dimensional faces of a simplex may be identified with each other.

Definition: Let $A,B$ be two indexing sets. A $\Delta$-complex (or delta complex) $X$ is
$$X = \left.\bigsqcup_{\alpha\in A} \Delta^{n_\alpha}_\alpha \right/\left\{\mathcal F_{\beta}^{k_\beta}\right\}_{\beta\in B}\ , \hspace{1cm} \mathcal F_\beta^{k_\beta} = \{\Delta_1^{k_\beta},\dots,\Delta_{m_\beta}^{k_\beta}\},$$
such that if $\sigma$ appears in the disjoint union, all of its lower dimensional faces also appear. The identification of the $k$-simplices in $\mathcal F^k$ is done in the natural (linear) way, and restricting to identified faces gives the identification of the $\mathcal F^{k-1}$ where the faces appear.

To define simplicial homology of a simplicial or $\Delta$-complex $X$, fix an ordering of the set of 0-simplices (which gives an ordering of every $\sigma\in X$), define $C_k$ to be the free abelian group generated by all $\sigma\in X$ of dimension $k$ (defined by $k+1$ 0-simplices), and define a boundary map
$$\begin{array}{r c l} \partial_k\ :\ C_k & \to & C_{k-1}, \\\ [v_0,\dots,v_k] & \mapsto & \sum_{i=0}^k(-1)^i[v_0,\dots,\widehat{v_i},\dots,v_k]. \end{array}$$
Then $H_k(X):= \text{ker}(\partial_k)/\text{im}(\partial_{k+1})$.

Recall the standard $n$-cell is $e^n = \{x\in \R^n\ :\ | x| \leqslant 1\}$, also known as the $n$-disk or $n$-ball.


Definition: Let $X_0$ be a finite set. A cell complex (or CW complex) is a collection $X_0,X_1,\dots$ where
$$X_k := \left.X_{k-1}\bigsqcup_{\alpha\in A_k} e^k_\alpha \right/\left\{\partial e^k_\alpha\sim f_{k,\alpha}(\dy e^k_\alpha)\right\}_{\alpha\in A_k},$$
where the $f_{k,\alpha}$ describe how to attach $k$-cells to the $(k-1)$-skeleton $X_{k-1}$, for $k\>1$. $X_k$ may also be described by pushing out $e^k\sqcup_{\dy e^k}X_{k-1}$. Note that $\dy e^k = S^{k-1}$, the $(k-1)$-sphere.

To define cellular homology, we need more tools (relative homology and excision) that require a blog post of their own.

References: Hatcher (Algebraic topology, Chapter 2.1)