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

The quotient is obtained from its image $Z\cong S^1$ by attaching the interior of $D^2$. The boundary map winds $n$ times around $Z$, so this is the cyclic Moore space in dimension one $M(\mathbb{Z}/n,1)$. Its cellular chain complex isTherefore

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.

At a point of the open two-cell, the local link is a circle, so local homology in degree two has rank one. At a point of $Z$, a neighborhood consists of $n$ half-discs sharing their boundary interval; its link is the Theta graph ${\Theta }_n$. The local homology from a link calculation givesWhen $n>2$, this local rank characterizes the points of $Z$. Since local homology is invariant under a homeomorphism, every homeomorphism satisfies $f(Z)=Z$, and in particular $f(Z)\subset Z$.

For $n=2$, the two pages form an ordinary disc neighborhood, and $Y$ is ${\mathbb{RP}}^2$. The distinguished circle $Z$ is a projective line, but a projective linear homeomorphism can carry it to a different projective line. Thus the conclusion does not hold.