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Floer homotopy theory and degenerate Lagrangian intersections

Kenneth Blakey

Abstract

We give a lower bound on the number of intersection points of a Lagrangian pair via Steenrod squares on Lagrangian Floer cohomology induced from a Floer homotopy type. The main technical input is a computation of the associated graded of the action-filtration of the Floer homotopy type in terms of Morse homotopy theory (precisely, Conley index theory). We also prove a lower bound using the quantum cap product on Lagrangian Floer cohomology.

Floer homotopy theory and degenerate Lagrangian intersections

Abstract

We give a lower bound on the number of intersection points of a Lagrangian pair via Steenrod squares on Lagrangian Floer cohomology induced from a Floer homotopy type. The main technical input is a computation of the associated graded of the action-filtration of the Floer homotopy type in terms of Morse homotopy theory (precisely, Conley index theory). We also prove a lower bound using the quantum cap product on Lagrangian Floer cohomology.

Paper Structure

This paper contains 32 sections, 49 theorems, 342 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Quantum cap-length
  4. Brief overview of Floer homotopy
  5. Main results
  6. Morse theory: homology
  7. Classical approach
  8. Functional-analytic approach
  9. Conley index theory
  10. Basics
  11. Gradient flow
  12. S-duality
  13. Co-H-spaces
  14. Isolated critical points
  15. Steenrod squares
  16. ...and 17 more sections

Key Result

Theorem 1.3

Suppose Assumption assu:overview and that $L_0$ intersects $L_1$ transversely, then we have the lower bound

Theorems & Definitions (103)

  • Conjecture 1.1: Arnol'd Conjecture
  • Remark 1.2
  • Theorem 1.3: Theorem 1 Flo88b
  • Theorem 1.4: Theorem 3 Hof88, Theorem 1 Flo89a
  • Theorem 1.5: Theorem 1.9 HP22
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10: CJS95,Lar21
  • ...and 93 more