publication . Article . Preprint . 2007

Kuranishi homology and Kuranishi cohomology

Joyce, Dominic;
  • Published: 24 Jul 2007
  • Country: United Kingdom
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 coh...
arXiv: Mathematics::Symplectic GeometryMathematics::Algebraic GeometryMathematics::Complex Variables
free text keywords: Mathematics - Symplectic Geometry
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(b) Let (U, Γ, φ), (V, Δ, ψ), p ∈ φ(U/Γ) ∩ ψ(V /Δ), (Up, Γp, φp), (Vp, Δp, ψp), ρp : Γp → Δp and σp : Up → Vp be as in Definition 2.9, and (U, Γ, φ), (V, Δ, ψ) lift to (EU , Γ, φˆ), (EV , Δ, ψˆ) on E with projections πU : EU → U , πV : EV → V as in (a). Set EUp = πU−1(Up), EVp = πV−1(Vp), so that EUp , EVp are vector bundles over Up, Vp with projections πU : EUp → Up,

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