Proper actions and supported-section-valued cohomology

TL;DR

This paper establishes a cohomological vanishing and isomorphism result for proper $Z^d$-actions on manifolds, linking cohomology with sections of sheaves and their descent to orbifold quotients.

Contribution

It generalizes cohomology vanishing and descent results for proper $Z^d$-actions on manifolds to broader sheaf contexts and supports, extending previous work.

Findings

Cohomology $H^p(Z^d, ext{sections})$ vanishes for $p eq d$

At $p=d$, cohomology equals sections of the descent sheaf

Results apply to sheaves with $ ext{support}$ conditions in a broad setting

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.