Nested cobordisms, Cyl-objects and Temperley-Lieb algebras

TL;DR

The paper develops a discrete cobordism framework for nested manifolds, culminating in a focused study of the striped cylinder category ${ m Cyl}$. It uses nested Morse theory and Cerf decompositions to produce a complete generators-and-relations presentation for ${ m Cyl}$, enabling explicit construction of ${ m Cyl}$-objects and their functors to other categories. It then connects these structures to affine and annular Temperley–Lieb algebras and to cyclic objects, and introduces a cylinder bar construction and a doubling operation to generate new algebraic models from dualizable objects. The work lays the groundwork for classifying 2D nested TQFTs and provides concrete algebraic tools for exploring gluing in nested cobordism settings, with potential cross-links to TL theory and cyclic homology-type constructions.

Abstract

We introduce a discrete cobordism category for nested manifolds and nested cobordisms between them. A variation of stratified Morse theory applies in this case, and yields generators for a general nested cobordism category. Restricting to a low-dimensional example of the ``striped cylinder'' cobordism category Cyl, we give a complete set of relations for the generators. With an eye towards the study of TQFTs defined on a nested cobordism category, we describe functors Cyl$\to\mathcal{C}$, which we call Cyl-objects in $\mathcal{C}$, and show that they are related to known algebraic structures such as Temperley-Lieb algebras and cyclic objects. We moreover define novel algebraic constructions inspired by the structure of Cyl-objects, namely a doubling construction on cyclic objects analogous to edgewise subdivision, and a cylindrical bar construction on self-dual objects in a monoidal category.

Figures (14)

• Figure 1: Generating cobordisms
• Figure 2: An example of a nested manifold $M_{0<1<2}$: a surface endowed with a 1-dimensional submanifold, which is itself endowed with a 0-dimensional submanifold.
• Figure 3: The figure on the left is an example of an individually Morse function which is not nested Morse. The figure on the right is nested Morse.
• Figure 4: Example of Morse data for the point $p$.
• Figure 5: Example of normal and tangential Morse data for a local picture with a critical point of index $1$ in the submanifold $M_1$ and no critical points on $M_2$; $N$ is the normal slice.

Theorems & Definitions (141)

• Theorem 1.1
• Theorem 1.2: \ref{['thm: gens of cyl']}, \ref{['thm: minimal list of generators']}, \ref{['cor: relns in cyl']}
• Corollary 1.3: \ref{['cor:cyl rep data']}
• Theorem 1.4: \ref{['thm:cyl and cyclic cat']}, \ref{['thm:inclusion of c2 cyc into cyl']}, \ref{['cor:cyl0 obj gives c2 cyclic']}
• Theorem 1.5: \ref{['cor:cyl object is affine TLA']}, \ref{['cor:cyl0 obj is annular TLA']}
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5