Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 114 / 1 / b

Suppose that $f$ is not surjective, and choose $y\notin f(Y)$. If $y$ lies in the open two-cell, then $Y\setminus \{y\}$ deformation retracts onto the target circle of the degree-$n$ attaching map, so its first integral homology is $\mathbb{Z}$. If $y\in Z$, the punctured space deformation retracts onto a finite graph and again has free abelian first homology. The factorizationtherefore makes $f_{\ast }:H_1(Y)\to H_1(Y)$ factor through a free abelian group. Every homomorphism from the finite group $H_1(Y)\cong \mathbb{Z}/n$ to a free abelian group is zero, contradicting the assumed surjectivity of $f_{\ast }$.

The converse is false. The radial coordinate descends to a surjection $r:Y\to [0,1]$. The finite CW complex $Y$ is a Peano continuum, so the Hahn-Mazurkiewicz theorem supplies a continuous surjection $g:[0,1]\to Y$. Then $g\circ r:Y\to Y$ is surjective, but its induced map on $H_1$ is zero because it factors through the contractible interval.