Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 114 / 1

Choose the intersection point $p=[({\theta }_0,{\theta }_1)]$ as the zero-cell. Let $a$ be the horizontal one-cell $S^1\times \{{\theta }_1\}$ and $b$ the vertical one-cell $\{{\theta }_0\}\times S^1$. The usual CW complex structure on the torus has one two-cell $c$ attached by the word $ab{a}^{-1}{b}^{-1}$, and the extra disk gives a two-cell $d$ attached along $b$ with degree one. Collapsing $a$ leaves one zero-cell $p$, one one-cell $b$, and the two two-cells $c,d$. The cellular boundary formula givesTherefore

The Excision theorem says that if $Z\subseteq A\subseteq X$ and $\overline{Z}\subseteq \mathrm{int}A$, then inclusion inducesIt lets us compute local homology in arbitrarily small neighborhoods.

Let $C\subseteq X$ be the image of $\{{\theta }_0\}\times S^1$. At a point of $X\setminus C$, a neighborhood is a disk, whose link is a circle. At a point of $C\setminus \{p\}$, three half-disks meet along their diameters, and the link is a Theta graph, with first homology ${\mathbb{Z}}^2$. At $p$, the two folds created by collapsing the horizontal circle give a dumbbell graph as link: two circles joined by an interval. Its first homology is again ${\mathbb{Z}}^2$. The local homology from a link therefore givesBecause local homology is invariant under a homeomorphism, every self-homeomorphism of $X$ preserves the rank-two locus $C$.

It must also preserve $p$. Indeed, $p$ is the unique point of $C$ whose sufficiently small local link is a dumbbell graph; every other point of $C$ has a theta-graph link. The connecting edge of a dumbbell graph is a bridge in a graph, whereas no edge of a theta graph is a bridge, so these local topological types are distinct.