  We show that the space $\mathcal{R}^{\mathrm{pRc}}(W_g^{2n})$ of metrics with
positive Ricci curvature on the manifold $W^{2n}_g := \sharp^g (S^n \times
S^n)$ has nontrivial rational homology if $n \not \equiv 3 \pmod 4$ and $g$ are
both sufficiently large. The same argument applies to
$\mathcal{R}^{\mathrm{pRc}}(W_g^{2n} \sharp N)$ provided that $N$ is spin and
$W_g^{2n} \sharp N$ admits a Ricci positive metric.

Algebraic Topology