Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 16 / 1 / 3 / Solution

The space is the mapping torus of the antipodal map $A:S^n\to S^n$. For $n\ge 1$, its mapping degree is $\delta =(-1{)}^{n+1}$: $A$ is the restriction of $-I$ on ${\mathbb{R}}^{n+1}$, whose determinant is $(-1{)}^{n+1}$. The Wang sequence givesThe kernel terms are free abelian, so these short exact sequences split as groups. On $H_0(S^n)$ the map is the identity, and on $H_n(S^n)$ it is multiplication by $\delta$. For $n\ge 2$ this givesAt $n=1$, both contributions to degree one are present, giving $H_0=H_2=\mathbb{Z}$ and $H_1={\mathbb{Z}}^2$, with all other groups zero. At $n=0$, the antipodal map exchanges the two points of $S^0$, and the two cylinder intervals join into a single circle; thus $H_0=H_1=\mathbb{Z}$ and higher groups vanish. These are the homology of the antipodal sphere mapping torus cases.