Nested cobordisms, Cyl-objects and Temperley-Lieb algebras

This paper introduces a discrete cobordism category for nested manifolds, derives its generators and relations via stratified Morse theory, and establishes a correspondence between functors on the "striped cylinder" subcategory and algebraic structures like Temperley-Lieb algebras and cyclic objects, while also proposing novel algebraic constructions such as a doubling mechanism and a cylindrical bar construction.

The Gist

Imagine you are a master chef trying to understand the structure of a very complex, multi-layered cake. Usually, when mathematicians study shapes (manifolds), they look at the whole cake. But in this paper, the authors are interested in "Nested Manifolds."

Think of a nested manifold not just as a cake, but as a Russian nesting doll or a set of concentric rings.

• You have the big outer ring (the cake itself).
• Inside that, you have a smaller ring (a layer of frosting).
• Inside that, maybe a tiny ring (a cherry on top).
• And inside that, perhaps a speck of sugar.

The rule is that every inner layer must be a "clean" shape sitting inside the next one, like a smaller circle inside a bigger circle.

The Big Idea: The "Striped Cylinder"

The authors want to study how these nested shapes can change into one another. They call this process a "Cobordism."

Imagine you have a nested shape at the bottom of a table (the "input") and a different nested shape at the top (the "output"). A cobordism is the 3D space connecting them.

• If you have a circle with a dot inside, and you want to turn it into a circle with two dots inside, the cobordism is the "tube" or "bridge" that connects them.
• Because the dots are nested inside the circle, the bridge itself must be a "nested bridge." It's a tube where the inner tube (connecting the dots) is embedded inside the outer tube (connecting the circles).

The authors focus on a specific, simplified version of this called the "Striped Cylinder."

• The Object: A circle with some dots (marks) on it.
• The Bridge: A cylinder (like a toilet paper roll) where the dots are connected by lines (stripes) running up the side.
• The Goal: To figure out exactly how to build any possible bridge between these striped cylinders using a small set of basic building blocks.

The LEGO Analogy: Generators and Relations

The paper's first major achievement is like finding the LEGO instructions for these bridges.

1. The Generators (The Bricks): The authors prove that you don't need infinite types of bridges. You only need four basic moves to build any complex bridge:

   • Identity: Just a straight tube (doing nothing).
   • Twist: Rotating the stripes so the dots swap places as you go up.
   • Birth: A new pair of dots appears out of nowhere on the cylinder (like a bubble popping up).
   • Death: A pair of dots disappears into the void (like a bubble popping).

2. The Relations (The Rules): Just because you have bricks doesn't mean you can build anything in any order. There are rules.

   • Example: If you "birth" a pair of dots and immediately "death" them right next to each other, they cancel out. It's like drawing a circle and then erasing it; you're back to where you started.
   • Example: If you twist the cylinder, the order in which you birth or kill dots changes. The paper maps out exactly how these moves interact.

They created a complete "dictionary" of these rules. This means if someone gives you a complicated, messy bridge, you can use these rules to simplify it down to a standard, unique form.

The "Cyl-Object": The Translator

Once they have the rules for the bridges, they ask: "What happens if we translate these bridges into math?"

This is where they introduce "Cyl-objects."
Imagine you have a machine (a functor) that takes these geometric bridges and turns them into algebraic equations or vector spaces.

• The "Circle with $n$ dots" becomes a specific object (like a vector space of dimension $n$).
• The "Twist" becomes a rotation operation.
• The "Birth/Death" becomes a way to shrink or expand the space.

The paper shows that if you follow the rules of the bridges, the resulting math machine is a very famous, well-studied structure called a Temperley-Lieb Algebra.

• Analogy: It's like discovering that the rules for how you can shuffle a deck of cards are exactly the same as the rules for how electrons move in a specific quantum physics model. The paper proves that "Nested Cobordisms" and "Temperley-Lieb Algebras" are two different languages describing the same underlying reality.

Why Does This Matter?

1. Topological Quantum Field Theories (TQFTs): In physics, TQFTs are models that describe how particles interact based purely on the shape of space, ignoring distance or time. By understanding these "nested" bridges, the authors are building a new toolkit for physicists to model complex quantum systems, like those involving "defects" or "strands" in space.
2. New Math Structures: They invented a new way to "double" existing mathematical structures (called the "Cyl-bar construction"). Think of it as taking a standard musical scale and creating a new, richer scale that includes all the possible harmonics and twists, allowing for more complex compositions.

Summary

In simple terms, this paper:

1. Defines a new way to look at shapes inside shapes (Nested Manifolds).
2. Solves the puzzle of how to build any connection between these shapes using just four basic moves (Birth, Death, Twist, Identity).
3. Proves that these geometric moves follow the exact same rules as famous algebraic structures used in physics and knot theory.
4. Creates new mathematical tools (Cyl-objects) that could help physicists and mathematicians understand complex systems where layers of structure interact.

It's a bridge between the messy, visual world of shapes and the clean, rigid world of algebra, showing that they are speaking the same language all along.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the construction and classification of discrete cobordism categories for nested manifolds. While classical cobordism theory studies manifolds and the cobordisms between them, and stratified Morse theory handles spaces with singularities, there was a lack of a unified discrete framework for nested manifolds (manifolds containing embedded submanifolds of lower dimensions, which may themselves contain further submanifolds).

The authors aim to:

1. Define a rigorous category of nested cobordisms ($Cob_I$).
2. Establish a "generators and relations" presentation for a specific, low-dimensional subcategory called Cyl (striped cylinders).
3. Characterize the functors from this category to an arbitrary category $\mathcal{C}$ (called Cyl-objects).
4. Connect these structures to known algebraic objects, specifically Temperley-Lieb algebras (affine and annular) and cyclic objects, thereby providing new algebraic models for Topological Quantum Field Theories (TQFTs) on nested structures.

2. Methodology

A. Nested Manifolds and Cobordisms

The authors define a nested $I$-manifold (where $I = (d_1 < \dots < d_n)$) as an ordered tuple $(M_{d_n}, \dots, M_{d_1})$ where each $M_{d_i}$ is a closed subset of $M_{d_{i+1}}$ diffeomorphic to a $d_i$-manifold.

• Cobordism: A cobordism between two nested manifolds is a nested manifold of dimension one higher, where the boundary consists of the disjoint union of the source and target.
• Category $Cob_I$: Objects are closed nested manifolds; morphisms are diffeomorphism classes of nested cobordisms. Composition is defined via pushouts (gluing along boundaries).

B. Nested Morse Theory

To analyze the structure of $Cob_I$, the authors develop a version of Morse theory for nested manifolds using Stratified Morse Theory (Goresky-MacPherson).

• They define a nested Morse function as a tuple of functions $(f_n, \dots, f_1)$ where each $f_i$ is a Morse function on $M_{d_i}$, and the critical points of all $f_i$ are distinct.
• Theorem: Every nested cobordism admits an "excellent" nested Morse function. This allows the decomposition of any cobordism into elementary cobordisms containing at most one critical point.
• Critical Points: The critical points are classified by their index $j$ on a specific stratum $M_{d_i}$. The local Morse data is shown to be the product of tangential Morse data (on the stratum) and normal Morse data (on the normal bundle of the stratum).

C. The Striped Cylinder Category ($Cyl$)

The authors restrict their study to the subcategory $Cyl$ (a quotient of $Cob_{1<2}$):

• Objects: Circles $S^1$ with $k$ marked points ($S^1_k$).
• Morphisms: Surfaces $S^1 \times [0,1]$ with 1-dimensional submanifolds (lines) connecting the marked points, modulo diffeomorphism and the relation that contractible circles are set to zero.
• Generators: Using the elementary cobordisms from Morse theory, they identify four types of generators:
  1. $id_k$: Identity.
  2. $tw_k$: Twist (Dehn twist) on the marked points.
  3. $b^i_k$: Birth (creation of a pair of points).
  4. $d^i_k$: Death (destruction of a pair of points).

D. Normal Forms and Relations

The authors prove that every morphism in $Cyl$ can be uniquely factored into a sequence of:

1. Deaths (Type I)
2. Twists/Bracelets (Type II)
3. Births (Type III)

By analyzing the combinatorics of these factorizations and using invariants (death index, birth index, twist number, bracelet number), they derive a complete set of relations (Theorem 3.14 and Corollary 3.16) governing these generators. These relations include standard "snake" relations, commutation rules for births/deaths, and interaction rules with twists.

3. Key Contributions and Results

A. Generators and Relations for $Cyl$

Theorem 1.2: The category $Cyl$ is generated by $id_k, tw_k, b^i_k, d^i_k$ subject to a specific set of relations. This provides a complete algebraic presentation of the category.

B. Characterization of Cyl-Objects

Corollary 1.3: A functor $F: Cyl \to \mathcal{C}$ (a Cyl-object) is determined by:

• A sequence of objects $c_n \in \mathcal{C}$.
• Isomorphisms $t_n: c_n \to c_n$ (cyclic action).
• Maps $d^i_n: c_n \to c_{n-2}$ and $s^j_n: c_n \to c_{n+2}$ (analogous to face and degeneracy maps).
• These must satisfy relations (i)–(vi) which generalize the axioms of cyclic objects (Connes' $\Lambda$) and simplicial objects ($\Delta$).

C. Connections to Temperley-Lieb Algebras

The paper establishes deep links between Cyl-objects and Temperley-Lieb algebras:

• Affine Temperley-Lieb ($T^a$): Every Cyl-object induces a representation of the affine Temperley-Lieb algebra. The category $Cyl$ is shown to be a quotient of the affine diagram category $D^a$ where Dehn twists are identified with the identity.
• Annular Temperley-Lieb ($A_{tl}$): The even subcategory $Cyl_0$ relates to the annular Temperley-Lieb algebra (defined by Jones and Penneys). The authors show that $Cyl_0$-objects are "shaded annular objects" where the shading constraint is relaxed.

D. Connection to Cyclic Objects and New Constructions

• Inclusion of Cyclic Category: There is an inclusion $\Lambda^{op} \hookrightarrow Cyl_0$. Thus, every Cyl-object restricts to a cyclic object.
• $\sqrt{\Lambda}$ Category: The authors introduce a new category $\sqrt{\Lambda}$ where the cyclic generator $t_n$ is replaced by a "square root" $\sqrt{t_n}$ such that $\sqrt{t_n}^{2(n+1)} = id$. They show $Cyl_0$ is isomorphic to a subcategory of $\sqrt{\Lambda}$.
• Doubling Construction: They define a "doubling" construction on simplicial objects that turns a cyclic object into a $\sqrt{\Lambda}$-object.
• Cyl-bar Construction: Inspired by the cyclic bar construction, they define a Cyl-bar construction ($BCyl_\bullet(X)$) for self-dual objects in a monoidal category. This construction produces a Cyl-object using evaluation and coevaluation maps, generalizing the cyclic bar complex.

4. Significance and Impact

1. New Algebraic Structures: The paper introduces novel algebraic structures (Cyl-objects, $\sqrt{\Lambda}$-objects, and the Cyl-bar construction) that extend the theory of cyclic and simplicial objects. These provide a framework for studying "square roots" of cyclic actions.
2. TQFTs on Nested Manifolds: By providing a generators-and-relations presentation of $Cyl$, the authors lay the groundwork for defining Topological Quantum Field Theories on nested cobordisms. This is significant for mathematical physics, particularly in modeling systems with defects or boundaries (e.g., Chern-Simons theory with defects).
3. Unification of Diagram Algebras: The work unifies the study of affine and annular Temperley-Lieb algebras under the umbrella of nested cobordism categories, clarifying the role of Dehn twists and contractible loops in these diagrammatic algebras.
4. Stratified Morse Theory Application: The successful application of stratified Morse theory to nested manifolds provides a robust tool for analyzing the topology of embedded structures, which has potential applications in configuration spaces and knot theory.

In summary, this paper constructs a rigorous bridge between the geometry of nested manifolds and abstract algebra, offering a complete presentation of a specific cobordism category and revealing its deep connections to Temperley-Lieb algebras and cyclic homology.