Manifolds with non-vanishing vector field and vast homology

Question

Let $n \ge 3$. Is there n-fold $M^n$ with both $\chi(M)=0$ and $\dim H_*(M,\mathbb{R}) \ge$ given number?

Answer 1

The product of two manifolds is a manifold. The Euler characteristic is multiplicative, so $\chi(M \times N) = \chi(M) \cdot \chi(N)$. Since $\chi(S^1) = 0$, it's sufficient to take the product of $S^1$ with an $(n-1)$-manifold that satisfies the homology dimension condition. For the latter, take the connected sum of enough copies of $\underbrace{S^1 \times \cdots \times S^1}_{(n-1) \text{ times}}$.