Homology operator

Redirect to:

• Cell maps

Definition

Given a cell map $f \colon K \rightarrow L$, the homology operator induced by f, $$f_* \colon H_k(K) \rightarrow H_k(L), {\rm \hspace{3pt} for \hspace{3pt} all \hspace{3pt}} k,$$ is a linear operator given by $$f_*([x]) = [f_k(x)],$$ where $$f_k \colon C_k(K) \rightarrow C_k(L)$$ is the chain map.

Why is the homology well defined?

Because the chain map preserves the boundary: $$\partial_k f_{k-1} = f_k \partial_k.$$ In other words, this diagram commutes: $$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{ccccc} \da{\partial_3} & & \da{\partial_3} \\ C_2(K) & \ra{g_2} & C_2(L) \\ \da{\partial_2} & & \da{\partial_2} \\ C_{1}(K) & \ra{g_{1}} & C_{1}(L)\\ \da{\partial_1} & & \da{\partial_1} \\ C_{0}(K) & \ra{g_{0}} & C_{0}(L)\\ \da{\partial_0} & & \da{\partial_0} \\ 0 & & 0\\ \end{array} $$

Examples

We've already carried out a few computations...

For dimension $k>0$:

• The identity: identity, $Id_*=Id.$
• The constant: zero, $f_*=0$.
• Self-maps of the circle:
  • flip: negative $f_*(c) = -c.$
  • turn: identity.
  • twice wound: doubling/stretch, $f_*(c) = 2c.$
• Collapse of the torus to its longitude: projection, $f_*(L) = 0, f_*(l) = c.$
• Inclusion of the circle in the plane: zero, $f_*(c) = 0.$

Exercise. Consider a few possible maps for each:

• inclusions of the circle into the torus;
• self-maps of the figure eight;
• inclusions of the circle to the sphere.

Properties

The following is obvious.

Theorem 1. The identity map induces the identity homology operator: $${\rm id}_{|K|} = {\rm id}_{H(K)}.$$

This is what we know about compositions of cell maps:

Theorem 2. The homology operator of the composition is the composition of the homology operators $$(gf)_* = g_*f_*.$$

The main result related to this theorem is below.

Theorem 3. Suppose K and L are cell complexes, if a map $$f \colon |K| \rightarrow |L|$$ is a cell map and a homeomorphism, and $$f^{-1} \colon |L| \rightarrow |K|$$ is a cell map too, then the homology operator $$f_* \colon H_k(K) \rightarrow H_k(L)$$ is an isomorphism for all $k$.

Proof. From the definition of inverse function, $$ff^{-1} = {\rm id}_{|L|},$$ $$f^{-1}f = {\rm id}_{|K|}.$$ From the above theorems, $$f_*(f^{-1})_* = {\rm id}_{H(L)},$$ $$(f^{-1})_* f_* = {\rm id}_{H(K)}.$$ $\blacksquare$