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Defining extended TQFTs via handle attachments

Benjamin Haïoun

TL;DR

The paper provides a finite presentation of the cobordism bicategory with corners via handle attachments and proves that a categorified $(n+1)$-TQFT together with coherent handle data on the standard handles, satisfying handle cancellation and $\iota$-invariance, uniquely extends to a once-extended $(n+2)$-TQFT. The construction relies on Morse data and Cerf theory adapted to cobordisms with corners to define a 2-functor on Hom-categories, which is then assembled into a symmetric monoidal 2-functor. A key outcome is that extensions are completely determined by the 0-handle value, enabling a clean classification of extensions and paving the way for skein-theoretic applications in non-semisimple settings. The approach unifies and extends prior frameworks (e.g., Juhász, Lurie, Walker) and provides a finite, practical handle-based route to fully extended TQFTs, with potential impact on locality, skein theories, and non-compact TQFTs.

Abstract

We give a finite presentation of the cobordism symmetric monoidal bicategory of (smooth, oriented) closed manifolds, cobordisms and cobordisms with corners as an extension of the bicategory of closed manifolds, cobordisms and diffeomorphisms. The generators are the standard handle attachments, and the relations are handle cancellations and invariance under reversing the orientation of the attaching spheres. In other words, given a categorified TQFT and 2-morphisms associated to the standard handles satisfying our relations, we construct a once extended TQFT.

Defining extended TQFTs via handle attachments

TL;DR

The paper provides a finite presentation of the cobordism bicategory with corners via handle attachments and proves that a categorified -TQFT together with coherent handle data on the standard handles, satisfying handle cancellation and -invariance, uniquely extends to a once-extended -TQFT. The construction relies on Morse data and Cerf theory adapted to cobordisms with corners to define a 2-functor on Hom-categories, which is then assembled into a symmetric monoidal 2-functor. A key outcome is that extensions are completely determined by the 0-handle value, enabling a clean classification of extensions and paving the way for skein-theoretic applications in non-semisimple settings. The approach unifies and extends prior frameworks (e.g., Juhász, Lurie, Walker) and provides a finite, practical handle-based route to fully extended TQFTs, with potential impact on locality, skein theories, and non-compact TQFTs.

Abstract

We give a finite presentation of the cobordism symmetric monoidal bicategory of (smooth, oriented) closed manifolds, cobordisms and cobordisms with corners as an extension of the bicategory of closed manifolds, cobordisms and diffeomorphisms. The generators are the standard handle attachments, and the relations are handle cancellations and invariance under reversing the orientation of the attaching spheres. In other words, given a categorified TQFT and 2-morphisms associated to the standard handles satisfying our relations, we construct a once extended TQFT.

Paper Structure

This paper contains 19 sections, 13 theorems, 62 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Outline
  4. The cobordism bicategory
  5. Handle attachments and main result
  6. Handle attachments
  7. Handle cancellation
  8. $\iota$-invariance
  9. Main result
  10. Handle decompositions for manifolds with corners
  11. A result on Hom categories
  12. Morse data and decompositions
  13. Existence and unicity of Morse data
  14. Inducing a 2-functor from handle attachment data
  15. Relation (2):
  16. ...and 4 more sections

Key Result

Theorem 3.8

Given a categorified TQFT $\mathcal{Z}^\varepsilon: \operatorname{\textsc{Cob}}_{n+1+\varepsilon}\to \mathcal{C}$, $n\geq 0$, and 2-morphisms such that there exists a unique $(n+1+1)$-TQFT extending $\mathcal{Z}^\varepsilon$ such that $\mathcal{Z}(H_k)=Z_k$. Reciprocally, given an $(n+1+1)$-TQFT $\mathcal{Z}$, taking $\mathcal{Z}^\varepsilon$ to be the restriction of $\mathcal{Z}$ to $\operator

Figures (4)

  • Figure 1: A cobordism with collared corners.
  • Figure 2: A Morse saddle $H_1$ in dimension 2 (so n=0). The vertical (or side) boundary is made of gradient lines for $f(x,y) = y^2-x^2$. The top and bottom boundaries are made of level sets for $f$.
  • Figure 3: The handle attachment along an embedded framed sphere for $n+2=2$ and $k=1$.
  • Figure 4: A Morse datum on a cobordism with corners.

Theorems & Definitions (49)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 39 more