Proper actions and supported-section-valued cohomology

Abstract

Consider a proper action of $\mathbb{Z}^d$ on a smooth (perhaps non-paracompact) manifold $M$. The $p^{th}$ cohomology $H^p(\mathbb{Z}^d,\ \Gamma_{\mathrm{c}}(\mathcal{F}))$ valued in the space of compactly-supported sections of a natural sheaf $\mathcal{F}$ on $M$ (such as those of smooth function germs, smooth $k$-form germs, etc.) vanishes for $p\ne d$ (the cohomological dimension of $\mathbb{Z}^d$) and, at $d$, equals the space of compactly-supported sections of the descent ($\mathbb{G}$-invariant push-forward) $\mathcal{F}/\mathbb{Z}^d$ to the orbifold quotient $M/\mathbb{Z}^d$. We prove this and analogous results on $\mathbb{Z}^d$ cohomology valued in $\Phi$-supported sections of an equivariant appropriately soft sheaf $\mathcal{F}$ in a broader context of $\mathbb{Z}^d$-actions proper with respect to a paracompactifying family of supports $\Phi$, in the sense that every member of $\Phi$ has a neighborhood small with respect to the action in Palais' sense.