Pushforward (homology)

In algebraic topology, the pushforward of a continuous function $f$ : $X\to Y$ between two topological spaces is a homomorphism $f_{*}:H_n\left(X\right)\to H_n\left(Y\right)$ between the homology groups for $n\ge 0$. 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 ${\sigma }_X$ : ${\mathrm{\Delta }}^n\to X$ with $f$ to obtain a singular n-simplex of $Y$, $f_{\mathrm{\#}}\left({\sigma }_X\right)=f{\sigma }_X$ : ${\mathrm{\Delta }}^n\to Y$. Then we extend $f_{\mathrm{\#}}$ linearly via $f_{\mathrm{\#}}\left(\sum _t{n}_t{\sigma }_t\right)=\sum _t{n}_t{f}_{\mathrm{\#}}\left({\sigma }_t\right)$.

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

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

Hence $f_{\mathrm{\#}}$ induces a homomorphism between the homology groups $f_{*}:H_n\left(X\right)\to H_n\left(Y\right)$ for $n\ge 0$.

Properties and homotopy invariance

Two basic properties of the push-forward are:

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

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

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

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

References

• Allen Hatcher, Algebraic topology. Cambridge University Press, ISBN 0-521-79160-X and ISBN 0-521-79540-0