Pushforward (homology)

In algebraic topology, the pushforward of a continuous function $$f$$ : $$X \rightarrow Y$$ between two topological spaces is a homomorphism $$f_{*}:H_n\left(X\right) \rightarrow H_n\left(Y\right)$$ between the homology groups for $$n \geq 0$$.

Homology is a functor which converts a topological space $$X$$ into a sequence of homology groups $$H_{n}\left(X\right)$$. (Often, the collection of all such groups is referred to using the notation $$H_{*}\left(X\right)$$; this collection has the structure of a graded ring.) In any category, a functor must induce a corresponding morphism. The pushforward is the morphism corresponding to the homology functor.

Definition for singular and simplicial homology
We build the pushforward homomorphism as follows (for singular or simplicial homology):

First we have an induced homomorphism between the singular or simplicial chain complex  $$C_n\left(X\right)$$ and $$C_n\left(Y\right)$$ defined by composing each singular n-simplex $$\sigma_X$$ : $$\Delta^n\rightarrow X$$ with $$f$$ to obtain a singular n-simplex of $$Y$$, $$f_{\#}\left(\sigma_X\right) = f\sigma_X$$ : $$\Delta^n\rightarrow Y$$. Then we extend $$f_{\#}$$ linearly via $$f_{\#}\left(\sum_tn_t\sigma_t\right) = \sum_tn_tf_{\#}\left(\sigma_t\right)$$.

The maps $$f_{\#}$$ : $$C_n\left(X\right)\rightarrow C_n\left(Y\right)$$ satisfy $$f_{\#}\partial = \partial f_{\#}$$ where $$\partial$$ is the boundary operator between chain groups, so $$\partial f_{\#}$$ defines a chain map.

We have that $$f_{\#}$$ takes cycles to cycles, since $$\partial \alpha = 0$$ implies $$\partial f_{\#}\left( \alpha \right) = f_{\#}\left(\partial \alpha \right) = 0$$. Also $$f_{\#}$$ takes boundaries to boundaries since $$ f_{\#}\left(\partial \beta \right) = \partial f_{\#}\left(\beta \right)$$.

Hence $$f_{\#}$$ induces a homomorphism between the homology groups $$f_{*} : H_n\left(X\right) \rightarrow H_n\left(Y\right)$$ for $$n\geq0$$.

Properties and homotopy invariance
Two basic properties of the push-forward are:


 * 1) $$\left( f\circ g\right)_{*} = f_{*}\circ g_{*}$$ for the composition of maps $$X\overset{g}{\rightarrow}Y\overset{f}{\rightarrow}Z$$.
 * 2) $$\left( \text{id}_X \right)_{*} = \text{id}$$ where $$\text{id}_X$$ : $$X\rightarrow X$$ refers to identity function of $$X$$ and  $$\text{id}\colon H_n\left(X\right) \rightarrow H_n\left(X\right)$$ refers to the identity isomorphism of homology groups.

A main result about the push-forward is the homotopy invariance: if two maps $$f,g\colon X\rightarrow Y$$ are homotopic, then they induce the same homomorphism $$f_{*} = g_{*}\colon H_n\left(X\right) \rightarrow H_n\left(Y\right)$$.

This immediately implies that the homology groups of homotopy equivalent spaces are isomorphic:

The maps $$f_{*}\colon H_n\left(X\right) \rightarrow H_n\left(Y\right)$$ induced by a homotopy equivalence $$f\colon X\rightarrow Y$$ are isomorphisms for all $$n$$.