Geometric chains of orbifolds with corners and singular homology

Abstract

We study a geometric homology theory (GHT for short). Its chains are generated by smooth maps from compact oriented orbifolds with corners to a smooth space, and two kinds of chains are set to zero: those admitting an orientation reversing automorphism, and degenerate ones, which, together with their boundaries, factor through smooth maps to orbifolds of lower dimension. Bisecting the source of a geometric chain along a hypersurface transverse to its corners changes the chain only by the boundary of an explicit prism and by prisms over its faces; cutting a geometric cycle along a regular level set of a smooth function on the target does not change its homology class. Geometric chains also behave well under products and pullbacks along fibre bundles, which overcomes some of the combinatorial rigidity of singular chains. We prove that geometric homology is naturally isomorphic to singular homology with rational coefficients for every smooth space admitting a good cover, such as smooth manifolds, orbifolds and locally finite polyhedra, and also for orbifolds with corners. The proof uses cutting operators along level sets of functions on the target, realized by explicit prisms, instead of triangulations. For continuous maps into arbitrary spaces we conjecture the same statement, and prove it for Euclidean neighbourhood retracts in degrees at most one, with surjectivity in degree two. Without the relation killing degenerate chains the comparison fails for orbifold chains: the weighted projective plane $\mathbb{P}(1,1,p)$, $p\ge5$ prime, mapped to a point is a cycle that does not bound. Applications on Morse theory and the BV structure on geometric chains of moduli spaces of bordered Riemann surfaces, following Costello and Harrelson--Voronov--Zúñiga, are given.