= Homology of the antipodal sphere mapping torus
{title2=$T_{-1}(S^n)$}
For $n\geq2$, the <Wang sequence> and the <mapping degree> $(-1)^{n+1}$ of the <antipodal map> give $H_0=H_1=\mathbb Z$. If $n$ is odd, $H_n=H_{n+1}=\mathbb Z$; if $n$ is even, $H_n=\mathbb Z/2$ and $H_{n+1}=0$. All other groups vanish. At $n=1$ the <mapping torus> is a torus with $H_1=\mathbb Z^2$. At $n=0$, the two-point swapping mapping torus is a circle. Every example is already a <closed manifold>; for $n\geq1$ it is orientable exactly when $n$ is odd.
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