= Solution
The space is the <mapping torus> of the <antipodal map> $A:S^n\to S^n$. For $n\geq1$, 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> gives
$$
0\longrightarrow\operatorname{coker}(1-A_*:H_j(S^n)\to H_j(S^n))\longrightarrow H_j(T_A)\longrightarrow\ker(1-A_*:H_{j-1}(S^n)\to H_{j-1}(S^n))\longrightarrow0.
$$
The 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 $n\geq2$ this gives
$$
\boxed{H_j(T_A;\mathbb Z)=\begin{cases}
\mathbb Z,&j=0,1,\\
\mathbb Z,&n\text{ odd and }j=n,n+1,\\
\mathbb Z/2,&n\text{ even and }j=n,\\
0,&\text{otherwise}.
\end{cases}}
$$
At $n=1$, 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.
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