Some new homology and cohomology theories of manifolds and orbifolds
For each manifold or effective orbifold $Y$ and commutative ring $R$, we define a new homology theory $MH_*(Y;R)$, $M$-$homology$, and a new cohomology theory $MH^*(Y;R)$, $M$-$cohomology$. For $MH_*(Y;R)$ the chain complex $(MC_*(Y;R),\partial)$ is generated by quadruples $[V,n,s,t]$ satisfying rel...
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| Format: | Journal article |
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2015
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| _version_ | 1797059126228942848 |
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| author | Joyce, D |
| author_facet | Joyce, D |
| author_sort | Joyce, D |
| collection | OXFORD |
| description | For each manifold or effective orbifold $Y$ and commutative ring $R$, we define a new homology theory $MH_*(Y;R)$, $M$-$homology$, and a new cohomology theory $MH^*(Y;R)$, $M$-$cohomology$. For $MH_*(Y;R)$ the chain complex $(MC_*(Y;R),\partial)$ is generated by quadruples $[V,n,s,t]$ satisfying relations, where $V$ is an oriented manifold with corners, $n\in\mathbb N$, and $s:V\to{\mathbb R}^n$, $t:V\to Y$ are smooth with $s$ proper near 0 in ${\mathbb R}^n$. We show that $MH_*(Y;R),MH^*(Y;R)$ satisfy the Eilenberg-Steenrod axioms, and so are canonically isomorphic to conventional (co)homology. The usual operations on (co)homology -- pushforwards $f_*$, pullbacks $f^*$, fundamental classes $[Y]$ for compact oriented $Y$, cup, cap and cross products $\cup,\cap,\times$ -- are all defined and well-behaved at the (co)chain level. Chains $MC_*(Y;R)$ form flabby cosheaves on $Y$, and cochains $MC^*(Y;R)$ form soft sheaves on $Y$, so they have good gluing properties. We also define $compactly$-$supported$ $M$-$cohomology$ $MH^*_{cs}(Y;R)$, $locally$ $finite$ $M$-$homology$ $MH_*^{lf}(Y;R)$ (a kind of Borel-Moore homology), and two variations on the entire theory, $rational$ $M$-($co$)$homology$ and $de$ $Rham$ $M$-($co$)$homology$. All of these are canonically isomorphic to the corresponding type of conventional (co)homology. The reason for doing this is that our M-(co)homology theories are very well behaved at the (co)chain level, and will be better than other (co)homology theories for some purposes, particularly in problems involving transversality. In a sequel we will construct virtual classes and virtual chains for Kuranishi spaces in M-(co)homology, with a view to applications of M-(co)homology in areas of Symplectic Geometry involving moduli spaces of $J$-holomorphic curves. |
| first_indexed | 2024-03-06T19:59:47Z |
| format | Journal article |
| id | oxford-uuid:26e04639-3970-4ada-a4da-28c3b290b52c |
| institution | University of Oxford |
| last_indexed | 2024-03-06T19:59:47Z |
| publishDate | 2015 |
| record_format | dspace |
| spelling | oxford-uuid:26e04639-3970-4ada-a4da-28c3b290b52c2022-03-26T12:03:36ZSome new homology and cohomology theories of manifolds and orbifoldsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:26e04639-3970-4ada-a4da-28c3b290b52cSymplectic Elements at Oxford2015Joyce, DFor each manifold or effective orbifold $Y$ and commutative ring $R$, we define a new homology theory $MH_*(Y;R)$, $M$-$homology$, and a new cohomology theory $MH^*(Y;R)$, $M$-$cohomology$. For $MH_*(Y;R)$ the chain complex $(MC_*(Y;R),\partial)$ is generated by quadruples $[V,n,s,t]$ satisfying relations, where $V$ is an oriented manifold with corners, $n\in\mathbb N$, and $s:V\to{\mathbb R}^n$, $t:V\to Y$ are smooth with $s$ proper near 0 in ${\mathbb R}^n$. We show that $MH_*(Y;R),MH^*(Y;R)$ satisfy the Eilenberg-Steenrod axioms, and so are canonically isomorphic to conventional (co)homology. The usual operations on (co)homology -- pushforwards $f_*$, pullbacks $f^*$, fundamental classes $[Y]$ for compact oriented $Y$, cup, cap and cross products $\cup,\cap,\times$ -- are all defined and well-behaved at the (co)chain level. Chains $MC_*(Y;R)$ form flabby cosheaves on $Y$, and cochains $MC^*(Y;R)$ form soft sheaves on $Y$, so they have good gluing properties. We also define $compactly$-$supported$ $M$-$cohomology$ $MH^*_{cs}(Y;R)$, $locally$ $finite$ $M$-$homology$ $MH_*^{lf}(Y;R)$ (a kind of Borel-Moore homology), and two variations on the entire theory, $rational$ $M$-($co$)$homology$ and $de$ $Rham$ $M$-($co$)$homology$. All of these are canonically isomorphic to the corresponding type of conventional (co)homology. The reason for doing this is that our M-(co)homology theories are very well behaved at the (co)chain level, and will be better than other (co)homology theories for some purposes, particularly in problems involving transversality. In a sequel we will construct virtual classes and virtual chains for Kuranishi spaces in M-(co)homology, with a view to applications of M-(co)homology in areas of Symplectic Geometry involving moduli spaces of $J$-holomorphic curves. |
| spellingShingle | Joyce, D Some new homology and cohomology theories of manifolds and orbifolds |
| title | Some new homology and cohomology theories of manifolds and orbifolds |
| title_full | Some new homology and cohomology theories of manifolds and orbifolds |
| title_fullStr | Some new homology and cohomology theories of manifolds and orbifolds |
| title_full_unstemmed | Some new homology and cohomology theories of manifolds and orbifolds |
| title_short | Some new homology and cohomology theories of manifolds and orbifolds |
| title_sort | some new homology and cohomology theories of manifolds and orbifolds |
| work_keys_str_mv | AT joyced somenewhomologyandcohomologytheoriesofmanifoldsandorbifolds |