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Equivariant Lagrangian Floer homology via cotangent bundles of $EG_N$

Guillem Cazassus

Abstract

We provide a construction of equivariant Lagrangian Floer homology $HF_G(L_0, L_1)$, for a compact Lie group $G$ acting on a symplectic manifold $M$ in a Hamiltonian fashion, and a pair of $G$-Lagrangian submanifolds $L_0, L_1 \subset M$. We do so by using symplectic homotopy quotients involving cotangent bundles of an approximation of $EG$. Our construction relies on Wehrheim and Woodward's theory of quilts, and the telescope construction. We show that these groups are independent in the auxiliary choices involved in their construction, and are $H^*(BG)$-bimodules. In the case when $L_0 = L_1$, we show that their chain complex $CF_G(L_0, L_1)$ is homotopy equivalent to the equivariant Morse complex of $L_0$. Furthermore, if zero is a regular value of the moment map $μ$ and if $G$ acts freely on $μ^{-1}(0)$, we construct two "Kirwan morphisms" from $CF_G(L_0, L_1)$ to $CF(L_0/G, L_1/G)$ (respectively from $CF(L_0/G, L_1/G)$ to $CF_G(L_0, L_1)$). Our construction applies to the exact and monotone settings, as well as in the setting of the extended moduli space of flat $SU(2)$-connections of a Riemann surface, considered in Manolescu and Woodward's work. Applied to the latter setting, our construction provides an equivariant symplectic side for the Atiyah-Floer conjecture.

Equivariant Lagrangian Floer homology via cotangent bundles of $EG_N$

Abstract

We provide a construction of equivariant Lagrangian Floer homology , for a compact Lie group acting on a symplectic manifold in a Hamiltonian fashion, and a pair of -Lagrangian submanifolds . We do so by using symplectic homotopy quotients involving cotangent bundles of an approximation of . Our construction relies on Wehrheim and Woodward's theory of quilts, and the telescope construction. We show that these groups are independent in the auxiliary choices involved in their construction, and are -bimodules. In the case when , we show that their chain complex is homotopy equivalent to the equivariant Morse complex of . Furthermore, if zero is a regular value of the moment map and if acts freely on , we construct two "Kirwan morphisms" from to (respectively from to ). Our construction applies to the exact and monotone settings, as well as in the setting of the extended moduli space of flat -connections of a Riemann surface, considered in Manolescu and Woodward's work. Applied to the latter setting, our construction provides an equivariant symplectic side for the Atiyah-Floer conjecture.
Paper Structure (28 sections, 29 theorems, 207 equations, 27 figures)

This paper contains 28 sections, 29 theorems, 207 equations, 27 figures.

Table of Contents

  1. Introduction
  2. Outline of the construction
  3. Statement of results
  4. Organization of the paper
  5. Telescopes
  6. Spaces
  7. Chain complexes
  8. Equivariant Morse homology
  9. Morse homology
  10. Pushforwards on Morse homology
  11. The equivariant Morse complex
  12. Equivariant Lagrangian Floer homology
  13. Hamiltonian actions
  14. Symplectic homotopy quotients
  15. Increment correspondences
  16. ...and 13 more sections

Key Result

Theorem 1

The construction outlined in Section ssec:outline_constr can be implemented in the exact and monotone setting (Assumptions ass:exact_setting, ass:monotone_setting): at the chain level, it defines a telescope where $CF_N$ is the Floer chain complex in $(M\times T^* EG_N )/\!\!/ G$ and $\alpha_N \colon CF_N \to CF_{N+1}$ chain morphisms induced by the inclusions $i_N\colon EG_N \to EG_{N+1}$. The ho

Figures (27)

  • Figure 1: Map from a product of telescopes to the telescope of products.
  • Figure 2: A grafted line.
  • Figure 3: Interior connected sum $u\#v$.
  • Figure 4: $u$ is homotopic to $\alpha \# \beta$.
  • Figure 5: The quilted surface $\underline{Z}$ and its decoration $(\underline{M}_N, \underline{L}_N)$.
  • ...and 22 more figures

Theorems & Definitions (96)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 2.1: Telescopes
  • Definition 2.2: Maps between telescopes
  • proof : Proof that the map is well-defined
  • Remark 2.3
  • Definition 2.4: Homotopies between telescopes
  • Proposition 2.5: Products
  • ...and 86 more