Some new homology and cohomology theories of manifolds and orbifolds
Abstract
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 EilenbergSteenrod 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 wellbehaved 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 BorelMoore 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2015
 DOI:
 10.48550/arXiv.1509.05672
 arXiv:
 arXiv:1509.05672
 Bibcode:
 2015arXiv150905672J
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Differential Geometry
 EPrint:
 232 pages, LaTeX