Proper actions and supported-section-valued cohomology
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909390935687168 |
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| author | Chirvasitu, Alexandru |
| author_facet | Chirvasitu, Alexandru |
| contents | Consider a proper action of $\mathbb{Z}^d$ on a smooth (perhaps non-paracompact) manifold $M$. The $p^{th}$ cohomology $H^p(\mathbb{Z}^d,\ Γ_{\mathrm{c}}(\mathcal{F}))$ 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 $Φ$-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 $Φ$, in the sense that every member of $Φ$ has a neighborhood small with respect to the action in Palais' sense. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_09857 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Proper actions and supported-section-valued cohomology Chirvasitu, Alexandru Group Theory Algebraic Topology General Topology K-Theory and Homology 20J06, 18G40, 55T10, 54D20, 54D15, 54B15, 55N30, 18F20 Consider a proper action of $\mathbb{Z}^d$ on a smooth (perhaps non-paracompact) manifold $M$. The $p^{th}$ cohomology $H^p(\mathbb{Z}^d,\ Γ_{\mathrm{c}}(\mathcal{F}))$ 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 $Φ$-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 $Φ$, in the sense that every member of $Φ$ has a neighborhood small with respect to the action in Palais' sense. |
| title | Proper actions and supported-section-valued cohomology |
| topic | Group Theory Algebraic Topology General Topology K-Theory and Homology 20J06, 18G40, 55T10, 54D20, 54D15, 54B15, 55N30, 18F20 |
| url | https://arxiv.org/abs/2411.09857 |