Pushforward (homology)

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

Homology is a functor which converts a topological space ${\displaystyle X}$ into a sequence of homology groups ${\displaystyle H_{n}\left(X\right)}$. (Often, the collection of all such groups is referred to using the notation ${\displaystyle H_{*}\left(X\right)}$; 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 ${\displaystyle C_{n}\left(X\right)}$ and ${\displaystyle C_{n}\left(Y\right)}$ defined by composing each singular n-simplex ${\displaystyle \sigma _{X}}$ : ${\displaystyle \Delta ^{n}\rightarrow X}$ with ${\displaystyle f}$ to obtain a singular n-simplex of ${\displaystyle Y}$, ${\displaystyle f_{\#}\left(\sigma _{X}\right)=f\sigma _{X}}$ : ${\displaystyle \Delta ^{n}\rightarrow Y}$. Then we extend ${\displaystyle f_{\#}}$ linearly via ${\displaystyle f_{\#}\left(\sum _{t}n_{t}\sigma _{t}\right)=\sum _{t}n_{t}f_{\#}\left(\sigma _{t}\right)}$.

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

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

Hence ${\displaystyle f_{\#}}$ induces a homomorphism between the homology groups ${\displaystyle f_{*}:H_{n}\left(X\right)\rightarrow H_{n}\left(Y\right)}$ for ${\displaystyle n\geq 0}$.

Properties and homotopy invariance

Two basic properties of the push-forward are:

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

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

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

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

References

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