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

First, the map ${\displaystyle f\colon X\to Y}$  induces a 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 ${\displaystyle \sigma _{X}\colon \Delta ^{n}\rightarrow X}$  with ${\displaystyle f}$  to obtain a singular n-simplex of ${\displaystyle Y}$ , ${\displaystyle f_{\#}\left(\sigma _{X}\right)=f\sigma _{X}\colon \Delta ^{n}\rightarrow Y}$ , and extending this linearly via ${\displaystyle f_{\#}\left(\sum _{t}n_{t}\sigma _{t}\right)=\sum _{t}n_{t}f_{\#}\left(\sigma _{t}\right)}$ .

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

Therefore, ${\displaystyle f_{\#}}$  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 ${\displaystyle n\geq 0}$ .

Two basic properties of the push-forward are:

1. ${\displaystyle \left(f\circ g\right)_{*}=f_{*}\circ g_{*}}$  for the composition of maps ${\displaystyle X{\overset {g}{\rightarrow }}Y{\overset {f}{\rightarrow }}Z}$ .
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.

(This shows the functoriality of the pushforward.)

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 (by the above properties) 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 ${\displaystyle n}$ .

• Pullback (cohomology)

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