Examples of odd dimensional rationally acyclic closed manifolds.

Question

Rationally acyclic mean that the all rational homology groups are vanish except the zeroth homology group. Here closed manifold mean compact without boundary. In even dimension, we have nice examples of rationally acyclic closed connected manifolds. For example even dimensional real projective spaces. The rational homology groups of $\mathbb{RP}^{2n}$ are $H_{0}(\mathbb{RP}^{2n};\mathbb{Q})=\mathbb{Q}$ and $H_{i>0}(\mathbb{RP}^{2n};\mathbb{Q})=0.$ It is possible for closed odd dimensional manifold or not? In oriented case it is not possible because the top homology group of oriented closed manifold is non zero.

Answer 1

No! Euler characteristics is an obstruction. Any closed odd dimensional manifold has $\chi=0$ [follows from Poincaré duality]. But if it is rationally acyclic, then $\chi=1$. Contradiction.