Pure and Applied Mathematics Quarterly

Volume 9 (2013)

Number 2

Special Issue: In Honor of Dennis Sullivan, Part 2 of 2

Lagrangian floer Theory over integers: spherically positive symplectic manifolds

Pages: 189 – 289

DOI: https://dx.doi.org/10.4310/PAMQ.2013.v9.n2.a1

Authors

Kenji Fukaya (Simons Center for Geometry and Physics, State University of New York, Stony Brook, N.Y., U.S.A.; Department of Mathematics, Kyoto University, Kyoto, Japan)

Yong-Geun Oh (Center for Geometry and Physics, Institute for Basic Sciences (IBS), Pohang, Gyungbuk, Korea; Department of Mathematics, POSTECH, Pohang, Korea; Department of Mathematics, University of Wisconsin, Madison Wisc., U.S.A.)

Hiroshi Ohta (Graduate School of Mathematics, Nagoya University, Nagoya, Japan; Korea Institute for Advanced Study, Seoul, Korea)

Kaoru Ono (Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan; Korea Institute for Advanced Study, Seoul, Korea)

Abstract

In this paper we study the Lagrangian Floer theory over $\mathbb{Z}$ or $\mathbb{Z}_2$. Under an appropriate assumption on ambient symplectic manifold, we show that the whole story of Lagrangian Floer theory in [6], [7] can be developed over $\mathbb{Z}_2$ coefficients, and over $\mathbb{Z}$ coefficients when Lagrangian submanifolds are relatively spin. The main technical tools used for the construction are the notion of the sheaf of groups, and stratification and compatibility of the normal cones applied to the Kuranishi structure of the moduli space of pseudo-holomorphic discs.

Keywords

Floer cohomology, Lagrangian submanifolds, orbifold, stack, stratified space, pseudo-holomorphic curve, spherically positive symplectic manifold

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Published 7 November 2013