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A contact homotopy type

Soham Chanda, Amanda Hirschi

TL;DR

The paper develops a comprehensive framework to lift contact homology into a Floer-homotopical setting by constructing global Kuranishi charts for genus-zero SFT moduli spaces and organizing them into a symmetric flow category with leveled, cogredient structures. It introduces base spaces via real-oriented and generalized blow-ups to account for domain degenerations and level formations, and builds perturbation data to obtain rel--$C^1$ global charts for both symplectizations and exact cobordisms. A stable, symmetric flow category with a stable complex lift is constructed, together with flow bimodules induced by exact cobordisms, yielding a robust, functorial, and potentially invariant higher-categorical refinement of contact homology. The framework places contact homology inside a stable $\,\infty$-category, enabling structured compositions, level-wise gluing, and naturality with respect to cobordisms, with explicit orientation data and a clear path to rational SFT via leveled and folded structures. This elevates the algebraic and homotopical understanding of SFT in the contact setting and provides new tools for computation and invariance questions in contact topology.

Abstract

Adapting the construction of global Kuranishi charts to the contact setting, we associate to any non-degenerate contact manifold a flow category based on Reeb orbits and moduli spaces of pseudo-holomorphic buildings. The construction lifts contact homology and is natural in the sense that to any exact symplectic cobordism we can associate a flow bimodule between the flow categories of its ends.

A contact homotopy type

TL;DR

The paper develops a comprehensive framework to lift contact homology into a Floer-homotopical setting by constructing global Kuranishi charts for genus-zero SFT moduli spaces and organizing them into a symmetric flow category with leveled, cogredient structures. It introduces base spaces via real-oriented and generalized blow-ups to account for domain degenerations and level formations, and builds perturbation data to obtain rel-- global charts for both symplectizations and exact cobordisms. A stable, symmetric flow category with a stable complex lift is constructed, together with flow bimodules induced by exact cobordisms, yielding a robust, functorial, and potentially invariant higher-categorical refinement of contact homology. The framework places contact homology inside a stable -category, enabling structured compositions, level-wise gluing, and naturality with respect to cobordisms, with explicit orientation data and a clear path to rational SFT via leveled and folded structures. This elevates the algebraic and homotopical understanding of SFT in the contact setting and provides new tools for computation and invariance questions in contact topology.

Abstract

Adapting the construction of global Kuranishi charts to the contact setting, we associate to any non-degenerate contact manifold a flow category based on Reeb orbits and moduli spaces of pseudo-holomorphic buildings. The construction lifts contact homology and is natural in the sense that to any exact symplectic cobordism we can associate a flow bimodule between the flow categories of its ends.

Paper Structure

This paper contains 53 sections, 76 theorems, 308 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Main results
  4. Background
  5. Buildings in symplectizations
  6. Buildings in exact symplectic cobordisms
  7. The global Kuranishi chart construction
  8. Base space
  9. Without levels
  10. Base space with levels
  11. Blow-up of $\mathcal{B}^{P}$
  12. Base space for disconnected domains
  13. Families of buildings
  14. Framings of buildings in symplectizations
  15. Determining unitary framings
  16. ...and 38 more sections

Key Result

Theorem A

Let $(Y,\xi)$ be a closed contact manifold equipped with a non-degenerate contact form $\lambda$. Suppose $J$ is a $\lambda$-adapted almost complex structure on $(Y,\xi)$ and $\Gamma^\pm$ are finite sequences of Reeb orbits.

Figures (7)

  • Figure 1: Morphisms of a tree
  • Figure 2: Three different level functions for an unleveled tree. We do not draw the trivial vertices for the sake of clarity; one can recover the trivial vertices uniquely by the differences of levels between adjacent vertices.
  • Figure 3: Morphisms of a cobordism tree
  • Figure 4: Corner blowup of $\mathbb{R}_+^2$.
  • Figure 5: Corner blowup of $\mathbb{R}_+^2$ corresponding to the tree with 3 vertices.
  • ...and 2 more figures

Theorems & Definitions (209)

  • Theorem A: Theorem \ref{['thm:leveled-gkc']}
  • Remark 1.1: Technical remark
  • Remark 1.2
  • Theorem B
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • Theorem C: Theorem \ref{['thm:flo_bim']}
  • Corollary 1.6: Lemma \ref{['lem:trivial-cobordism-diagonal-bimodule']}
  • Definition 2.1
  • ...and 199 more