Connected odd dimensional Manifold with all its even homology groups are trivial except the zeroth homology group.

Question

I need the examples of such connected odd dimensional manifolds whose even homology groups are trivial except zeroth homology group. Well known examples are odd dimensional spheres.

Answer 1

If $M$ is a closed connected $n$-dimensional manifold with $n = 2k + 1$ and $H_i(M; \mathbb{Z}) = 0$ for $i$ even, then in particular $H_{n-1}(M; \mathbb{Z}) = 0$ from which it follows that $M$ is orientable and hence $H_n(M;\mathbb{Z}) \cong \mathbb{Z}$.

Note that

\begin{align*} \chi(M) &= \sum_{i=0}^n(-1)^i\operatorname{rank}H_i(M; \mathbb{Z})\\ &= \sum_{i=0}^k\operatorname{rank} H_{2i}(M;\mathbb{Z}) - \sum_{i=0}^k\operatorname{rank} H_{2i+1}(M; \mathbb{Z})\\ &= 1 - \sum_{i=0}^{k-1}\operatorname{rank}H_{2i+1}(M; \mathbb{Z}) - 1\\ &= \sum_{i=0}^{k-1}\operatorname{rank}H_{2i+1}(M; \mathbb{Z}). \end{align*}

As $\chi(M) = 0$ for closed odd-dimensional manifolds, we see that $\operatorname{rank}H_{2i+1}(M; \mathbb{Z}) = 0$ for $i = 0, \dots, k-1$ so $H_{2i+1}(M; \mathbb{Z})$ must be torsion for $i = 0, \dots, k-1$. It follows that all such manifolds are rational homology spheres. Note however that not all odd-dimensional rational homology spheres will meet your requirements. For example, the Wu manifold $SU(3)/SO(3)$ is a rational homology sphere which has second integral cohomology group $\mathbb{Z}_2$; see here.

Aside from integral homology spheres, lens spaces provide examples, in particular, $\mathbb{RP}^n$ is an example of such a manifold. Note, one can create more examples by taking connected sums of two such manifolds.