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 $f_{{\#}}\left(\sum _{t}n_{t}\sigma _{t}\right)=\sum _{t}n_{t}f_{{\#}}\left(\sigma _{t}\right)$ .

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 $n\geq 0$ .

Properties and homotopy invariance

Main article: singular homology § Homotopy invariance

Two basic properties of the push-forward are:

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

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

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

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

References

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

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