Stated skein modules of 3-manifolds and TQFT

TL;DR

The paper develops a comprehensive framework for stated skein modules of marked 3-manifolds, extending classical Kauffman bracket skein modules to boundary markings that include intervals and circles. It establishes nontrivial root-of-unity phenomena causing non injectivity of natural gluing, cutting, and Frobenius-type maps, and introduces a robust TQFT-like structure by viewing stated skein modules as a symmetric monoidal functor to Morita categories, yielding Van Kampen type results and Heegaard-based computations. A central technical advancement is the identification of a Hochschild viewpoint, showing that splitting along a strict surface yields an isomorphism with Hochschild 0th homology and enabling triangle sums and modular decompositions. The work thus connects combinatorial skein theory with categorical TQFT, Hochschild homology, and quantum group representations, providing both explicit algebraic tools and topological interpretations for 3-manifolds with marked boundaries.

Abstract

We study the behaviour of the Kauffman bracket skein modules of 3-manifolds under gluing along surfaces. For this purpose we extend the notion of Kauffman bracket skein modules to $3$-manifolds with marking consisting of open intervals and circles in the boundary. The new module is called the stated skein module. The first main results concern non-injectivity of certain natural maps defined when forming connected sums along a sphere or along a closed disk. These maps are injective for surfaces, or for generic quantum parameter, but we show that in general they are not injective when the quantum parameter is a root of 1. The result applies to the classical skein modules as well. A particular interesting result is that when the quantum parameter is a root of 1, the empty skein is zero in a connected sum where each constituent manifold has non-empty marking. We also prove various non injectivity results for the Chebyshev-Frobenius map and the natural map induced by the deletion of marked balls. We then consider the general case of gluing along a surface, showing that the stated skein module can be interpreted as a monoidal symmetric functor from a category of "decorated cobordisms" to a Morita category of algebras and their bimodules. We apply this result to deduce several properties of stated skein modules as a Van-Kampen like theorem as well as a computation through Heegaard decompositions and a relation to Hochshild homology for trivial circle bundles over surfaces.

Figures (9)

• Figure 1: A half-ideal slit breaking $c$, with an interior ideal point.
• Figure 2: Moves M1 and M2 for isotopy of $D$
• Figure 3: Left: slitting along a properly embedding arc. Right: the equivalence relation. All circular bold arcs might be in the same component of $\partial \mathfrak S$.
• Figure 4: Left: Cutting along an interior ideal arc $c$. Right: The map $\mathsf{Cut}_c$. The case when $c$ is a circle is similar.
• Figure 5:

Theorems & Definitions (91)

• Theorem 1: Special case of Theorem \ref{['thm.connected3']}
• Theorem 2: See Theorem \ref{['thm.Frobenius']}
• Theorem 3: See Theorem \ref{['thm.CH']}
• Theorem 4: Special case of Theorem \ref{['thm.zero']}
• Theorem 5: See Theorem \ref{['teo:cutting']} for more precise statement
• Theorem 6: See Theorem \ref{['thm.split3']}
• Theorem 7: See Theorem \ref{['thm.tqft']}
• Definition 2.1
• Remark 2.2
• Definition 2.3