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Short overview
Overview
The spaces that we study will be made of elementary pieces with "trivial" topology.
As an example, a cubical complex has been understood as a collection of cubical cells on the grid ${\bf Z}^n$ so that the boundary operator is well defined.
Now we use a more general approach as cells of all shapes are allowed. The approach is also a more abstract as we separate the complex from the space in which it resides. This allows us to make the structures purely combinatorial and, therefore, amenable to computation.
Alternatively, we can have no redundancy but only for "nice" spaces.
Suppose $Q_k,k=0,1,2,...,$ are finite. Then we proceed inductively.
The $0$-skeleton $K^{(0)}$ is defined as the disjoint union of $0$-cells, as points: $$K^{(0)} = Q_0.$$
Suppose that the $m$-skeleton $K^{(m)}$ has been constructed, as a topological space. Now the $(m+1)$-skeleton $K^{(m+1)}$ is constructed as follows.
Suppose for each $(m+1)$-cell $a$ there is a gluing map $$f_a \colon \partial a \rightarrow K^{(m)},$$ where $\partial a$ is the boundary of $a$ (homeomorphic to the sphere ${\bf S}^{m-1}$). Then all $(m+1)$-cells are "attached" to $K^{(m)}$ by means of these maps resulting in the $(m+1)$-skeleton $K^{(m+1)}$. More precisely, it is defined as the disjoint union of the $m$-skeleton $K^{(m)}$ and all the $(m+1)$-cells, under a certain equivalence relation: $$K^{(m+1)} = (K^{(m)} \cup Q_{m+1})/ _{\sim},$$ where $\sim$ is given by:
Our complex has now a topological counterpart, called its realization: $$|K|=\bigcup _m K^{(m)}=\bigcup _k \bigcup _{\sigma \in Q_k} \text{im } \sigma.$$ We still have our gluing maps: $$f_a \colon \partial a \rightarrow K,$$ The images of the cells under these maps can also be called cells.
The gluing maps have to satisfy this condition:
The topology of the cell complex is understood via the equivalence of the following three statements about a subset $A$:
- it is open/closed,
- its preimage under the gluing map is open/closed,
- its intersection with each cell is open/close in that cell.
Combinatorially, cell complex $K$ is the combination of
- the collection of cells $K=\bigcup _m Q_m$,
- the skeleta $\{K^{(0)},K^{(1)},..\}$, and
- the collection of the gluing maps $\{f_a \colon \partial a \rightarrow K^{(n)},a \in K\}$.
Another term commonly used is "CW-complex".
Transformations of the spaces given by cell complexes are captured by cell-to-cell maps.
Given two cell complexes $K$ and $L$ and a function between them $$f:K \rightarrow L,$$ we assume that the dimensions of cells can't go up: $$a\in Q_m, f(a)\in Q_k \Rightarrow m \ge k,$$ and that the boundaries are preserved $$b\in \partial a \Rightarrow f(b) \in \partial f(a).$$ Then we call $f$ a cell map.
The last requirement is a discrete/combinatorial analogue of continuity.
Note: Recall that a continuous function $g$ is called a cell map if $$g(K^{(m)}) \subset L^{(m)}.$$
In order to be able to study what is going on in our spaces, we have to circumvent the lack any algebraic structure.
Recall $R$ is a ring.
Next, $k$-chains are formal linear combinations of $k$-cells with coefficients in $R$: $$C_k(K)=< Q_k(K) >=\left\{ \sum _i r_i a_i:r_i\in R,a_i\in Q_k(K) \right\}, k=0,1,....$$ It is a free module. It is given by its (commonly finite) basis, $Q_k(K)$.
The complete collection of chains is also a graded module, denoted by $$C(K)=\bigoplus _i C_i(K),$$ with basis $\cup Q_k(K)$. One can go back and forth between these two interpretations.
This module is equipped with boundary operators (denoted by the same letter $\partial$ as the boundary of a cell): $$\partial _k: C_k(K) \rightarrow C_{k-1}(K),k=0,1,....$$ One can think of it as a single, graded (of degree $-1$) operator: $$\partial =\bigoplus _k \partial _k: C(K) \rightarrow C(K).$$ One can go back and forth between these two interpretations.
When the chains come from a cell complex as above, then for every $k$-cell $a\in K$ any of its boundary $(k-1)$-cells $b\in \partial a$ is also in $K$. The sum of those is defined to be the boundary chain of this cell: $$\partial _k (a)=\sum _{b \in \partial a} b.$$
However, in general any sequence of linear operators that satisfy $$\partial _{k-1} \partial _k =0 : C_k(K) \rightarrow C_{k-2}(K)$$ can be used, below. Alternatively, we simply write $$\partial \partial =0.$$
The combination of these modules and linear operators $(C(K), \partial )$ is called the chain complex of $K$ over $R$.
Chain complexes may come from sources other than cell complexes. For example, all formal linear combinations of all singular cells in a topological space $X$: $$\sigma _k: {\bf B}^k \rightarrow X,$$ form one, the singular chain complex of $X$.
Note: The linear combinations can be taken over any ring $R$. However, the Universal Coefficient Theorem indicates that the homology over any ring can be derived from the one over the integers, but generally not vice versa. This suggests that the default choice should be $R={\bf Z}$.
Transformations of the spaces given by chain complexes are captured by linear operators.
If we extend a cell map $f:K\rightarrow L$ from the cells by linearity to the chains, the result is a chain operator, i.e., a linear map between chain complexes $$f \colon ( C(K), \partial ^K) \rightarrow (C(L), \partial ^L)$$ that preserves boundaries. In other words, it commutes with the boundary operator: $$f_{k-1} \partial_k^K = \partial_k^L f_k,$$ or simply: $$ f \partial = \partial f .$$ This requirement is an algebraic analogue of continuity.
Consider the commutative diagram (the commutativity is given by this preservation of boundary condition): $$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} \newcommand{\la}[1]{\!\!\!\!\!\!\!\xleftarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\ua}[1]{\left\uparrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{ccccccc} \da{\partial_3^K} & & \da{\partial_3^L} \\ C_2(K) & \ra{f_2} & C_2(L) \\ \da{\partial_2^K} & & \da{\partial_2^L} \\ C_{1}(K) & \ra{f_{1}} & C_{1}(L)\\ \da{\partial_1^K} & & \da{\partial_1^L} \\ C_{0}(K) & \ra{f_{0}} & C_{0}(L)\\ \da{\partial_0^K} & & \da{\partial_0^L} \\ 0& &0\\ \end{array} $$
Chain maps may come from sources other than cell maps. For example, a chain map $$g:Z\rightarrow Z$$ defined on the standard cubical complex $Z$ of the real line by $$g(\{0\})=\{0\},g(\{1\})=\{3\},$$ $$g([0,1])=[0,1]+[1,2]+[2,3],$$ may represent motion of an object from point $0$ to point $3$ over the period of time from $0$ to $1$. Further, any chain map $$g:Z\rightarrow C$$ to any chain complex $C$ is a parametric curve and $$g:Z\times Z \rightarrow C$$ is a parametric surface, etc.
