2015/09/18 by Dominic Joyce, Joyce, Dominic · 1 citation
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.1509.05672
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),∂) is generated by quadruples [V,n,s,t] satisfying relations, where V is an oriented manifold with corners, n∈\mathbb N, and s:V→\mathbb Rn, t:V→ Y are smooth with s proper near 0 in \mathbb Rn. 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 ∪,∩,× -- 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.