A symmetric monoidal Frohman-Nicas TQFT for sutured manifolds

TL;DR

This paper generalizes the Frohman-Nicas TQFT to sutured manifolds, providing a symmetric monoidal functor framework that extends previous theories and relates to $ ext{Spin}^c$ structures.

Contribution

It reinterprets and extends the Frohman-Nicas TQFT for sutured manifolds, removing connectivity restrictions and establishing a symmetric monoidal functor structure.

Findings

Decategorification of bordered sutured Heegaard Floer homology is used.

The TQFT is extended to arbitrary cobordisms with no connectivity restrictions.

Relation established between decategorified theory and $ ext{Spin}^c$ structures.

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.