Some new homology and cohomology theories of manifolds and orbifolds
Some new homology and cohomology theories of manifolds and orbifolds
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流形和轨道的一些新同调和上同调理论
DOI：
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
D. Joyce
中科院分区：
文献类型：
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作者：
D. Joyce
对于每一个流形或有效轨道$Y$和交换环$R$，我们定义了一个新的同调理论$MH_*(Y；R)$, $M$-$同调$，和一个新的上同调理论$MH^*(Y；R)$, $M$-$上同调$。对于$MH_*(Y；R)$，链复形$（MC_*(Y；R)，偏）$是由满足关系的四元组$[V,n,s,t]$生成的，其中$V$是一个有角的有向流形，$ninmathbb n$, $s:V o{mathbb R}^n$, $t:V o Y$是光滑的，$s$在${mathbb R}^n$中近似于0。
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, $ninmathbb N$, and $s:V o{mathbb R}^n$, $t:V o Y$ 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 $cup,cap, imes$ -- 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.
影响因子：
1.7
作者：
Joyce D
通讯作者：
Joyce D
在有角的流形上
DOI：
10.48550/arxiv.0910.3518
发表时间：
2009
期刊：
arXiv e-prints
影响因子：
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作者：
Joyce Dominic
通讯作者：
Joyce Dominic
仓西空间的新定义
DOI：
10.48550/arxiv.1409.6908
发表时间：
2014
期刊：
arXiv e-prints
影响因子：
--
作者：
Joyce Dominic
通讯作者：
Joyce Dominic