Free actions of large finite abelian groups on weakly Lefschetz cohomologically symplectic manifolds force total Betti number at least 2^k, with a torus-like structure theorem for elementary abelian p-groups.
Free actions on products of real projective spaces
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We prove that if $G=(\mathbb{Z}/2)^r$ acts freely and cellularly on a finite-dimensional CW-complex $X$ homotopy equivalent to $\mathbb{R}P ^{n_1} \times \cdots \times \mathbb{R} P ^{n_k}$ with trivial action on the mod-$2$ cohomology, then $r \leq \mu (n_1)+ \cdots + \mu(n_k )$ where for each integer $n\geq 0$, $\mu (n)=0$ if $n$ is even, $\mu(n)=1$ if $n\equiv 1$ mod 4, and $\mu(n)=2$ if $n\equiv 3$ mod 4. This proves a homotopy-theoretic version of a conjecture of Cusick.
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Finite abelian group actions on weakly Lefschetz cohomologically symplectic manifolds
Free actions of large finite abelian groups on weakly Lefschetz cohomologically symplectic manifolds force total Betti number at least 2^k, with a torus-like structure theorem for elementary abelian p-groups.