Kuranishi homology and Kuranishi cohomology: a User's Guide

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. This paper is a brief introduction to the author's book arXiv:0707.3572. Let Y be an orbifold and R a Q-algebra. We define the Kuranishi homology KH_*(Y;R) of Y with coefficients in 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 KH_*(Y;R) is isomorphic to singular homology. We define a Poincare dual theory of Kuranishi cohomology KH^*(Y;R), isomorphic to compactly-supported cohomology, using a cochain complex KC^*(Y;R) spanned over R by [X,f,C], for X a compact Kuranishi space with corners, f : X --> Y a submersion, and C "co-gauge-fixing data". We also define simpler theories of Kuranishi bordism KB_*(Y;R) and Kuranishi cobordism KB^*(Y;R), for R a commutative ring. These 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.