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A symmetric monoidal Frohman-Nicas TQFT for sutured manifolds

Andrew Manion, Elijah Rutter

TL;DR

This work constructs a symmetric monoidal sutured Frohman–Nicas TQFT by decategorifying bordered sutured invariants and interpreting the resulting maps on 3d sutured cobordisms as a functor into graded abelian groups modulo sign. It resolves half-projectivity issues that arise in the classical FN TQFT and extends functoriality to disconnected inputs/outputs within the domain $ ext{Cob}^{ ext{sut}}_{2+1}$, targeting $ ext{Ab}^{ ext{Z-gr}}_{\pm 1}$. It further connects the Spin$^c$-refined decategorified invariant to Florens–Massuyeau’s Alexander framework via an Alexander function $ extsf{A}_{ullet}$ over $bZ[G]$ or $bQ[H]$, enabling Alexander-theoretic interpretations of knot complements from decategorified bordered sutured data. Finally, the construction recovers the ordinary FN TQFT on connected cobordisms and opens avenues toward extended and non-semisimple TQFT perspectives within a sutured setting.

Abstract

By analyzing the decategorification of bordered sutured Heegaard Floer homology, we reinterpret and generalize the classical Frohman-Nicas TQFT for the Alexander polynomial in the setting of 3d sutured cobordisms between sutured surfaces. In this setting, the Frohman-Nicas TQFT maps for arbitrary cobordisms between surfaces, with no connectivity restrictions, get interpreted as part of an honest symmetric monoidal functor (under disjoint union) with no half-projectivity zeroes. We also relate the decategorified bordered sutured theory with $\mathrm{Spin}^c$ structures to a sutured version of Florens-Massuyeau's $G$-analogue of the Frohman-Nicas TQFT.

A symmetric monoidal Frohman-Nicas TQFT for sutured manifolds

TL;DR

This work constructs a symmetric monoidal sutured Frohman–Nicas TQFT by decategorifying bordered sutured invariants and interpreting the resulting maps on 3d sutured cobordisms as a functor into graded abelian groups modulo sign. It resolves half-projectivity issues that arise in the classical FN TQFT and extends functoriality to disconnected inputs/outputs within the domain , targeting . It further connects the Spin-refined decategorified invariant to Florens–Massuyeau’s Alexander framework via an Alexander function over or , enabling Alexander-theoretic interpretations of knot complements from decategorified bordered sutured data. Finally, the construction recovers the ordinary FN TQFT on connected cobordisms and opens avenues toward extended and non-semisimple TQFT perspectives within a sutured setting.

Abstract

By analyzing the decategorification of bordered sutured Heegaard Floer homology, we reinterpret and generalize the classical Frohman-Nicas TQFT for the Alexander polynomial in the setting of 3d sutured cobordisms between sutured surfaces. In this setting, the Frohman-Nicas TQFT maps for arbitrary cobordisms between surfaces, with no connectivity restrictions, get interpreted as part of an honest symmetric monoidal functor (under disjoint union) with no half-projectivity zeroes. We also relate the decategorified bordered sutured theory with structures to a sutured version of Florens-Massuyeau's -analogue of the Frohman-Nicas TQFT.
Paper Structure (7 sections, 12 theorems, 37 equations, 10 figures)

This paper contains 7 sections, 12 theorems, 37 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Arc diagrams and sutured surfaces
  4. Heegaard diagrams and sutured cobordisms
  5. CW decompositions
  6. Decompositions from Heegaard diagrams
  7. Orientations of cells

Key Result

Theorem 1.7

If $H_2(Y,R^-) = 0$, consider the up-to-sign element note that if $\operatorname{rank} \ker(i_*) < K$ (i.e. $H_2(Y,R^+) \neq 0$) then $|K_{Y,\Gamma}| = 0$, while if $\operatorname{rank} \ker(i_*) = K$ (i.e. $H_2(Y,R^+) = 0$) then $\wedge^K \ker(i_*)$ can be viewed as an up-to-sign element rather than a subspace. If $H_2(Y,R^-) \neq 0$, let $|K_{Y,\Gam where $\varepsilon$ is defined as in Definiti

Figures (10)

  • Figure 1: A prototypical example where the half-projectivity zero appears in the Frohman--Nicas TQFT.
  • Figure 2: Left to right: An $\alpha$-arc diagram $\mathcal{Z}$; the sutured surface $F(\mathcal{Z})$ associated to $\mathcal{Z}$ (we draw $S^+$ in orange and $S^-$ in purple); $S_-$ with belt spheres indicated and boundary orientation reversed; the $\beta$-arc diagram $\mathcal{Z}^*$ dual to $\mathcal{Z}$.
  • Figure 3: Top: an arc diagram for a genus $g$ sutured surface $(F,\Lambda)$ with no $S^+$ intervals. Bottom: an arc diagram for a genus $g$ sutured surface $(F,\Lambda)$ with $\left(\sum_{i=1}^r n_i\right)$-many $S^+$ intervals on $r$ mixed boundary circles. To conserve space, the above arc diagrams have been rotated $90^{\circ}$ clockwise from the conventions under Definition \ref{['def:ArcDiagram']}.
  • Figure 4: An $\alpha$-$\alpha$ bordered sutured Heegaard diagram $\mathcal{H}$ from $\mathcal{Z}_0$ to $\mathcal{Z}_1$. The subset $\psi(\mathbf{Z}_1 \sqcup -\mathbf{Z}_0) \subset \partial \Sigma$ is drawn in black, and $\partial_{\mathrm{non-gluing}}(\Sigma) \subset \partial \Sigma$ is drawn in grey. The pairs of circles (also drawn in black) labeled $A$, $B$, and $C$ should be glued together to form cylinders; see Figure \ref{['fig:CobordismOrientations']} for a more 3-dimensional picture of such a cylinder. We draw $\Sigma$ with the standard orientation as a region of $\mathbb{R}^2$, consistent with Figure \ref{['fig:CobordismOrientations']} and Remark \ref{['rem:ChangeOfConventions']}. With this choice, the arc diagrams are drawn pointing downward rather than upward as in the conventions below Definition \ref{['def:ArcDiagram']}; this is purely a matter of presentation, since arc diagrams have no preferred planar direction.
  • Figure 5: Left: portion of a bordered sutured Heegaard diagram $\mathcal{H}$ with Heegaard surface $\Sigma$. Right: outgoing or $\mathbf{Z}_1$ side of the sutured cobordism $Y(\mathcal{H})$; the front rectangular face is $\mathbf{Z}_1 \times [0,1]$. By convention we will usually shade $R^+$ in grey, but this convention may be relaxed in more intricate figures such as this one.
  • ...and 5 more figures

Theorems & Definitions (61)

  • Remark 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Example 1.5
  • Remark 1.6
  • Theorem 1.7
  • Example 1.8
  • Lemma 1.9
  • Proposition 1.10
  • ...and 51 more