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Reduced symplectic homology and string topology

Kai Cieliebak, Alexandru Oancea

TL;DR

This work develops a unified algebraic framework in which the fundamental string topology operations, the loop product and loop coproduct, live on a common, reduced domain. By introducing reduced loop (and symplectic) homology and carefully controlling continuation-data dependencies via secondary continuation maps, the authors construct a commutative cocommutative unital infinitesimal anti-symmetric bialgebra structure on reduced loop homology, with a Sullivan-type identity augmented by an extra term that vanishes under natural topological hypotheses. The theory is carried through in the symplectic setting for strongly $R$-essential Weinstein domains, with a cone/Rabinowitz-Floer perspective that yields coassociativity, anti-symmetry, and compatibilities under open/closed string dualities, including Lagrangian (open-string) analogues. Applications to string topology follow via Viterbo-type isomorphisms, yielding explicit bialgebra structures for reduced loop homology and for odd-dimensional spheres, and revealing a rich interplay between Floer theory, Rabinowitz Floer homology, and loop space topology. Overall, the paper provides a robust computational and conceptual toolkit for combining loop product and coproduct in a controlled, homotopy-coherent algebraic framework with broad implications for open-closed string theories and topological field theories.

Abstract

We introduce a common domain of definition for the loop product and the loop coproduct, reduced loop homology, on which they combine to a unital infinitesimal anti-symmetric bialgebra structure. In particular, a relation conjectured by Sullivan holds with an extra term. The structure depends on choices governed by secondary continuation maps. These results on string topology are proved in the more general context of reduced symplectic homology for a suitable class of Weinstein manifolds.

Reduced symplectic homology and string topology

TL;DR

This work develops a unified algebraic framework in which the fundamental string topology operations, the loop product and loop coproduct, live on a common, reduced domain. By introducing reduced loop (and symplectic) homology and carefully controlling continuation-data dependencies via secondary continuation maps, the authors construct a commutative cocommutative unital infinitesimal anti-symmetric bialgebra structure on reduced loop homology, with a Sullivan-type identity augmented by an extra term that vanishes under natural topological hypotheses. The theory is carried through in the symplectic setting for strongly -essential Weinstein domains, with a cone/Rabinowitz-Floer perspective that yields coassociativity, anti-symmetry, and compatibilities under open/closed string dualities, including Lagrangian (open-string) analogues. Applications to string topology follow via Viterbo-type isomorphisms, yielding explicit bialgebra structures for reduced loop homology and for odd-dimensional spheres, and revealing a rich interplay between Floer theory, Rabinowitz Floer homology, and loop space topology. Overall, the paper provides a robust computational and conceptual toolkit for combining loop product and coproduct in a controlled, homotopy-coherent algebraic framework with broad implications for open-closed string theories and topological field theories.

Abstract

We introduce a common domain of definition for the loop product and the loop coproduct, reduced loop homology, on which they combine to a unital infinitesimal anti-symmetric bialgebra structure. In particular, a relation conjectured by Sullivan holds with an extra term. The structure depends on choices governed by secondary continuation maps. These results on string topology are proved in the more general context of reduced symplectic homology for a suitable class of Weinstein manifolds.
Paper Structure (35 sections, 42 theorems, 179 equations, 11 figures)

This paper contains 35 sections, 42 theorems, 179 equations, 11 figures.

Table of Contents

  1. Introduction
  2. $R$-essential Morse functions
  3. Two flavors of reduced symplectic homology
  4. Reduced symplectic homology $\overline{SH}_*(W)$
  5. Symplectic homology relative to the continuation map $SH_*(W;\mathrm{im}\, c)$
  6. Comparison between $\overline{SH}_*(W)$ and $SH_*(W;\mathrm{im}\, c)$
  7. Coproducts on $SH_*(W;\mathrm{im}\, c)$
  8. Coproduct defined by a choice of continuation data
  9. Secondary continuation map
  10. Dependence of the coproduct on the choice of continuation data
  11. Strongly $R$-essential Weinstein domains
  12. Bialgebra structure on reduced symplectic homology
  13. Unital infinitesimal anti-symmetric bialgebras
  14. Bialgebra structure on reduced symplectic homology
  15. Coassociativity
  16. ...and 20 more sections

Key Result

Theorem 1.1

Assume $\dim M= 1$ or $\dim M \ge 3$. The loop product on ${\mathbb{H}}_*\Lambda$ descends to $\overline{\mathbb{H}}_*\Lambda$ and the loop coproduct on ${\mathbb{H}}_*(\Lambda,\Lambda_0)$ extends to $\overline{\mathbb{H}}_*\Lambda$ (canonically if $H_1M=0$). Each such extension $\lambda$ defines to where $1$ denotes the identity map and $\eta$ the unit for the product $\mu$.

Figures (11)

  • Figure 1: Definition of the coproduct on $SH_*(W;\mathrm{im}\, c)$.
  • Figure 2: The operation ${\boldsymbol{\beta}}_\mathcal{D}$.
  • Figure 3: Secondary continuation map from coproduct.
  • Figure 4: The relation ${\boldsymbol{\lambda}}_{\mathcal{D}'} = {\boldsymbol{\lambda}}_\mathcal{D} + ({\boldsymbol{\mu}}\otimes 1)(1\otimes {\boldsymbol{c}}_{\mathcal{D},\mathcal{D}',\mathcal{C}})-(1\otimes{\boldsymbol{\mu}})(\tau{\boldsymbol{c}}_{\mathcal{D},\mathcal{D}',\mathcal{C}}\otimes 1)$.
  • Figure 5: The product ${\boldsymbol{\mu}}$ and the coproduct ${\boldsymbol{\lambda}}$.
  • ...and 6 more figures

Theorems & Definitions (110)

  • Theorem 1.1
  • Proposition 1.2
  • proof
  • proof
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Lemma 2.5
  • ...and 100 more