Kuranishi homology and Kuranishi cohomology

TL;DR

This paper introduces Kuranishi homology and cohomology theories for moduli spaces in symplectic geometry, providing new tools that simplify virtual cycle construction and potentially prove integrality conjectures.

Contribution

It defines new Kuranishi (co)homology and bordism theories, establishing their isomorphism with classical homology and cohomology, and introduces Gromov-Witten invariants in Kuranishi bordism.

Findings

Kuranishi homology is isomorphic to singular homology.

Kuranishi cohomology is isomorphic to compactly-supported cohomology.

New Gromov-Witten type invariants in Kuranishi bordism are introduced.

Abstract

A Kuranishi space is a topological space with a Kuranishi structure, defined by Fukaya and Ono. Kuranishi structures occur naturally on moduli spaces of J-holomorphic curves in symplectic geometry. Let Y be an orbifold and R a commutative ring or Q-algebra. We define two kinds of Kuranishi homology KH_*(Y;R). The chain complex KC_*(Y;R) defining KH_*(Y;R) is spanned over R by [X,f,G], for X a compact oriented Kuranishi space with corners, f : X --> Y smooth, and G "gauge-fixing data" which makes Aut(X,f,G) finite. Our main result is that these are isomorphic to singular homology. We define Poincare dual Kuranishi cohomology, isomorphic to compactly-supported cohomology. We define five kinds of Kuranishi (co)bordism spanned by isomorphism classes[X,f] for X a compact oriented Kuranishi space without boundary and f : X --> Y smooth. They are new topological invariants, and we show they are very large. These theories are powerful new tools in symplectic geometry. Defining virtual cycles and chains for moduli spaces of J-holomorphic curves is trivial in Kuranishi (co)homology. There is no need to perturb moduli spaces, and no problems with transversality. This gives major simplifications in Lagrangian Floer cohomology. We define new Gromov-Witten type invariants in Kuranishi bordism, over Z not Q. We sketch how these may be used to prove the integrality conjecture for Gopakumar-Vafa invariants. This paper is surveyed in arXiv:0710.5634.