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Multiplicative structures on cones and duality

Kai Cieliebak, Alexandru Oancea

Abstract

We initiate the study of multiplicative structures on cones and show that cones of Floer continuation maps fit naturally in this framework. We apply this to give a new description of the multiplicative structure on Rabinowitz Floer homology and cohomology, and to give a new proof of the Poincaré duality theorem which relates the two. The underlying algebraic structure admits two incarnations, both new, which we study and compare: on the one hand the structure of $A_2^+$-algebra on the space $\mathcal{A}$ of Floer chains, and on the other hand the structure of $A_2$-algebra involving $\mathcal{A}$, its dual $\mathcal{A}^\vee$ and a continuation map from $\mathcal{A}^\vee$ to $\mathcal{A}$.

Multiplicative structures on cones and duality

Abstract

We initiate the study of multiplicative structures on cones and show that cones of Floer continuation maps fit naturally in this framework. We apply this to give a new description of the multiplicative structure on Rabinowitz Floer homology and cohomology, and to give a new proof of the Poincaré duality theorem which relates the two. The underlying algebraic structure admits two incarnations, both new, which we study and compare: on the one hand the structure of -algebra on the space of Floer chains, and on the other hand the structure of -algebra involving , its dual and a continuation map from to .

Paper Structure

This paper contains 31 sections, 26 theorems, 279 equations, 43 figures.

Table of Contents

  1. Introduction
  2. Products on cones
  3. The point of view of split pairs of $A_\infty$-algebras
  4. The point of view of $A_\infty$-bimodules
  5. $A_2$-triples
  6. Morphisms of $A_2$-triples
  7. Examples
  8. Cones of Floer continuation maps
  9. Closed symplectically aspherical manifolds
  10. Completions of Liouville domains
  11. Rabinowitz Floer homology and cohomology rings via cones
  12. The Rabinowitz Floer homology ring
  13. The Rabinowitz Floer cohomology ring
  14. The unit in cohomology
  15. Poincaré duality
  16. ...and 16 more sections

Key Result

Theorem 2.6

Given a homotopy retraction of pairs as above, and given an $A_\infty$-algebra structure on $\mathcal{B}$ such that $\mathcal{A}$ is a subalgebra, there is an $A_\infty$-structure on $\mathcal{B}'$ such that $\mathcal{A}'$ is a subalgebra, and there are extensions of $P,I$ to $A_\infty$-morphisms of

Figures (43)

  • Figure 1: The families of Floer continuation maps $\{K_\lambda\}$, $\mathcal{H}_\vee$ and $\mathcal{H}_\wedge$.
  • Figure 2: Trees for the transferred map $\sigma'$.
  • Figure 3: Curves defining the maps $\mu$, $m_L$, $m_R$ in Floer theory.
  • Figure 4: Curves defining the maps $\sigma$, $\tau_L$, $\tau_R$ in Floer theory.
  • Figure 5: Curves defining the map $\beta$ in Floer theory.
  • ...and 38 more figures

Theorems & Definitions (81)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Homotopy transfer for split pairs
  • Proposition 2.7
  • Definition 2.8
  • Definition 2.9
  • Lemma 2.10
  • ...and 71 more