This site is being phased out.

The algebra of chains

From Mathematics Is A Science
(Redirected from Chain group)
Redirect page
Jump to navigationJump to search

Redirect to:

Here is a more advanced, group theory (and linear algebra) based, approach to chain complexes and homology theory of cell complexes.

The first step is to add more structure to the complex - orientation of cells. An orientation of a cell is simply a specific choice of the order of its vertices. We illustrate this idea below - with the cell complex representation of the circle:

Orientation of cells in circle.jpg

Here the $1$-cells aren't just edges $a, b, c$ but $a = AB, b = CB, c = AC$. We could have chosen any other order of vertices: $a = BA, b = CB, c = CA$, etc.

Another choice of cells' orientations will produce a different algebra... but the same topology!

A chain previously was defined as simply a "combination" of cells. In the above example, there are only these $1$-chains: $$0, a, b, c, a + b, b + c, c + a, a + b + c. $$

This makes sense geometrically, but this time we want to capture many other possible ways to go around the circle: going twice, or thrice around it, or going in the opposite direction.

A chain is then a "formal" linear combination of finitely many oriented cells, such as $3a + 5b - 17c$.

We will concentrate on integral chains, i.e., ones with integer coefficients. From this point of view, the chains previously discussed are binary. Then the former will result in "homology over ${\bf Z}$" and the latter in "homology over ${\bf Z}_2$". Other choices are possible as well: ${\bf R}$, ${\bf Q}$, etc or any ring (we need the ring structure to define the algebraic operations on chains). The general theory is presented in Homology theory.

More formally, for a given cell complex $K$ let $C_k(K)$ denote the set of all integral chains: $$C_k(K) = \left\{ \displaystyle\sum_{i}s_i \sigma_i \colon s_i \in {\bf Z}, \sigma_i {\rm \hspace{3pt} is \hspace{3pt} a \hspace{3pt}} k{\rm -cell \hspace{3pt} in \hspace{3pt}} K \right\}.$$

It is easy to define addition of chains by assigning coefficients of each cell: $$A = \displaystyle\sum_i s_i \sigma_i,$$ $$B = \displaystyle\sum_i t_i \sigma_i,$$ then $$A + B = \displaystyle\sum_i(s_i + t_i) \sigma_i.$$

To see that the operation above is well defined, one can assume that each sum lists all cells present in both sums -- some with zero coefficients.

Theorem. $C_k(K)$ is a group with respect to chain addition.

Proof. Verify the axioms of groups.

(1) Identity element: $$0 = \displaystyle\sum_i 0 \cdot \sigma_i.$$

(2) Inverse element: $$A = \displaystyle\sum_i s_i \sigma_i, {\rm \hspace{3pt} then}$$ $$-A = \displaystyle\sum_i (-s_i)\sigma_i.$$

(3) Associativity: $$\begin{array}{l} A + (B + C) &= \displaystyle\sum_i (s_i + t_i + u_i) \sigma_i \\ &= (A + B) + C. \end{array}$$ $\blacksquare$

Theorem. $C_k(K)$ is an abelian group generated by the $k$-cells of $K$.

In particular, in the example above of a triangle representing the circle we have

  • $C_0(K) = <A> = {\bf Z},$
  • $C_1(K) = <a,b,c> = {\bf Z} \times {\bf Z} \times {\bf Z},$
  • $C_2(K) = 0$, etc.

Note: Until a relation is established between cells/chains of different dimensions, this algebra can't capture the topology of the cell complex. This relation is given by the boundary operator.

Further see Homology in dimension 1.