Pushforward (homology)

Let $X$ and $Y$ be two topological spaces and $f:X\; ightarrow\; Y$ a continuous function. Then $f$ induces a homomorphism between the homology groups $f\_\{*\}:H\_nleft(X\; ight)\; ightarrow\; H\_nleft(Y\; ight)$ for $ngeq0$. We say that $f\_\{*\}$ is the pushforward induced by $f$.

Definition for singular and simplicial homology

We build the pushforward homomorphism as follows, for singular or simplicial homology:

First we have a induced homomorphism between the singular or simplicial chain complex $C\_nleft(X\; ight)$ and $C\_nleft(Y\; ight)$ defined by composing each singular n-simplex $sigma\_X:Delta^n\; ightarrow\; X$ with $f$ to obtain a singular n-simplex of $Y$, $f\_\{\#\}left(sigma\_X\; ight)=fsigma\_X:Delta^n\; ightarrow\; Y$. Then we extend $f\_\{\#\}$ linearly via $f\_\{\#\}left(sum\_tn\_tsigma\_t\; ight)\; =\; sum\_tn\_tf\_\{\#\}left(sigma\_t\; ight)$.

The maps $f\_\{\#\}:C\_nleft(X\; ight)\; ightarrow\; C\_nleft(Y\; ight)$ 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 $partial\; f\_\{\#\}left(\; alpha\; ight)\; =\; f\_\{\#\}left(partial\; alpha\; ight)\; =\; 0$. Also $f\_\{\#\}$ takes boundaries to boundaries since .

Hence $f\_\{\#\}$ induces a homomorphism between the homology groups $f\_\{*\}:H\_nleft(X\; ight)\; ightarrow\; H\_nleft(Y\; ight)$ for $ngeq0$.

Properties and homotopy invariance

Two basic properties of the push-forward are:

# $left(\; fcirc\; g\; ight)\_\{*\}\; =\; f\_\{*\}circ\; g\_\{*\}$ for the composition of maps $Xoverset\{f\}\{\; ightarrow\}Yoverset\{g\}\{\; ightarrow\}Z$.
# $left(\; id\_X\; ight)\_\{*\}\; =\; id$ where $id\_X:X\; ightarrow\; X$ refers to identity function of $X$ and $id:H\_nleft(X\; ight)\; ightarrow\; H\_nleft(X\; ight)$ refers to the identity isomorphism of homology groups.

A main result about the push-forward is the homotopy invariance: if two maps $f,g:X\; ightarrow\; Y$ are homotopic, then they induce the same homomorphism $f\_\{*\}\; =\; g\_\{*\}:H\_nleft(X\; ight)\; ightarrow\; H\_nleft(Y\; ight)$.

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

The maps $f\_\{*\}:H\_nleft(X\; ight)\; ightarrow\; H\_nleft(Y\; ight)$ induced by a homotopy equivalence $f:X\; ightarrow\; Y$ are isomorphisms for all $n$.

References

* Allen Hatcher, [http://www.math.cornell.edu/~hatcher/AT/ATpage.html "Algebraic topology."] Cambridge University Press, ISBN 0-521-79160-X and ISBN 0-521-79540-0