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(infinity,n)-category of cobordisms in nLab

Contents

under construction

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Manifolds and Cobordisms

manifolds and cobordisms

cobordism theory, Introduction

Definitions

Genera and invariants

Classification

Theorems

Cobordism theory

cobordism theory = manifolds and cobordisms + stable homotopy theory/higher category theory

Concepts of cobordism theory

Functorial Quantum Field Theory

Contents

Idea

nn-Dimensional manifolds (possibly and usually equipped with certain structure, notably for instance with orientation, framing-structure or more general G-structure) should naturally form an (∞,n)-category of extended cobordisms whose

  • objects are 0-dimensional (oriented) manifolds (disjoint unions of (oriented) points);

  • 1-morphisms are (oriented) cobordisms between disjoint unions of (oriented) points;

  • 2-morphisms are cobordisms between 1-dimensional cobordisms

  • etc.

  • (n+1)-morphisms are diffeomorphisms between nn-dimensional cobordisms;

  • (n+2)-morphisms are smooth homotopies of these;

  • etc.

The (∞,n)(\infinity,n)-category of cobordisms is the subject of the cobordism hypothesis.

Definition

As an nn-fold complete Segal space

Here is an outline of the idea of the definition of Bord (∞,n)Bord_{(\infty,n)} as given in (Lurie) where the main point, apart from the (∞,n)-category machinery in the background, is definition 2.2.9.

The idea is to start with thinking of nn-dimensional cobordisms as forming something like an n-fold category by simply saying that the collection of composites of cobordisms is given by big cobordisms with markings on them, indicating where we think of them as being composed.

Let’s first do this for composition in one direction, as in an ordinary 1-category of nn-dimensional cobordisms.

Consider a manifold X↪V×ℝX \hookrightarrow V \times \mathbb{R} embedded in a vector space of the form V×ℝV \times \mathbb{R}. We can think of this as a manifold canonically equipped with a coordinate function ϕ:X↪V×ℝ→ℝ\phi : X \hookrightarrow V \times \mathbb{R} \to \mathbb{R} that measures the “height” or maybe better the “length” of the embedded manifold.

We can pick a bunch of numbers {t j∈ℝ}\{t_j \in \mathbb{R}\} and think of these as marking a bunch of slices of XX, the preimages ϕ −1(t j)\phi^{-1}(t_j). We can think of these slices as being the (n−1)(n-1)-dimensional boundary manifolds at which a sequence of manifolds have been glued together to produce XX.

(there is an obvious picture to be drawn and uploaded here, maybe somebody finds the time and energy)

In this way an embedded manifold X↪V×ℝX \hookrightarrow V \times \mathbb{R} and a set of kk-numbers {t i}\{t_i\} may represent an element in the space of sequences of composable cobordisms. To make this work as expected, the markings on XX may not be too irregular, so we should impose some conditions on what qualifies as a marked manifold. The precise statement is given further below.

The collection of these tuples, consisting of an embedded manifold X↪V×ℝX \hookrightarrow V \times \mathbb{R} and a collection of kk numbers {t i∈ℝ}\{t_i \in \mathbb{R}\} naturally form a simplicial set, which is like the nerve of the 1-category of nn-dimensional cobordisms under composition in one direction.

To generalize this from just a 1-categorical structure to an nn-categorical structure, we simply take a manifold XX as before, but now draw markings on it in nn transversal directions, thereby putting a kind of grid on it that subdivides the manifold into cubical slices. A manifold with such subdivision on it may then be regarded as giving an element in the space of nn-dimensional pasting diagrams in an nn-fold category.

To formalize this more general case, we embed XX not just into a V×ℝV \times \mathbb{R}, but a V×ℝ nV \times \mathbb{R}^n. This then provides us with nn different coordinate functions ϕ i:X↪V×ℝ n→p iℝ\phi_i : X \hookrightarrow V \times \mathbb{R}^n \stackrel{p_i}{\to} \mathbb{R} on XX, each running along one of the directions in which we may think of XX as having been glued from smaller manifolds.

A collection of markings indicating such gluing is now a collection of numbers {t j 1},{t j 2},⋯{t j n}\{t_j^1\}, \;\{t_j^2\}, \; \cdots \{t_j^n\}, one for each of these directions.

For each direction this yields a simplicial set of such structures, to be thought of as the nerve of the category of cobordisms under composition in one of these directions. Taken together this is an nn-fold simplicial set

Δ op×Δ op×⋯×Δ op→Set \Delta^{op} \times \Delta^{op} \times \cdots \times \Delta^{op} \to Set

which is like the nerve of an nn-fold category of cobordisms.

When suitable regularity conditions are imposed on this data, there is naturally a topology on each of these sets of embedded marked cobordisms, that makes this into an nn-fold simplicial topological space

Δ op×Δ op×⋯×Δ op→Top. \Delta^{op} \times \Delta^{op} \times \cdots \times \Delta^{op} \to Top \,.

To get rid of the dependence of this construction on VV, we can let VV “grow arbitrarily large” by taking the colimit of the above nn-fold cosimplicial spaces as VV ranges over the finite dimensional subspaces of ℝ ∞\mathbb{R}^\infty.

The resulting nn-fold simplicial topological space obtained by this colimit then is essentially the (∞,n)-category Bord nBord_n that we are after. It turns out that it actually is an nn-fold Segal space. We just formally complete it to an n-fold complete Segal space

Bord n:(Δ op) n→Top. Bord_n : (\Delta^{op})^n \to Top \,.

This, then, is a model for the (∞,n)-category of extended nn-dimensional cobordisms.

As a blob nn-category

There is a definition of a blob n-category of nn-cobordisms. See there for more details.

Examples

Bord 2 frBord_2^{fr}

Some comments on 2-framed 2-cobordisms.

Consider the pictures in (Schommer-Pries 13, figure 5).

Layer 1 image/svg+xml positive point negative point ev: pt+ u pt- -> 0 coev ev^R ε η Serre automorphism

Somebody should produce pictures like this here…

Well, it worked in the preview; a little help?

Let γ\gamma be a 1-dimensional manifold of the form of the interval [0,1][0,1]. A 2-framing of γ\gamma is a trivialization of Tγ⊕ℝT\gamma \oplus \mathbb{R}. Let {1}⊂ℝ\{1\} \subset \mathbb{R} be the canonical basis of ℝ\mathbb{R}. If we think of the plane ℝ 2\mathbb{R}^2 as equipped with its canonical 2-framing, then a 2-framing of γ\gamma is induced by embedding γ\gamma into the plane and shading one of its two sides. This identifies at each point x∈γx \in \gamma the tangent space to γ\gamma at that point with the tangent vector to the embedding of γ\gamma as a vector in ℝ 2\mathbb{R}^2 and identifies 1∈ℝ1\in \mathbb{R} with the vector in ℝ 2\mathbb{R}^2 orthogonal to this tangent vector and pointing into the shaded region.

This shows that if γ\gamma is regarded with its two endpoints both as incoming or both as outgoing, then the induced 2-framing of these endpoints is opposite to each other. This way such an arc is a morphism from the union of the “positive point” and the “negative point” to the empty 0-manifold, hence is a unit/counit exhibiting these as dual objects.

Properties

Adjoints

Bord nBord_n is an (∞,n)-category with all adjoints.

Relation to Thom spectrum

For n→∞n \to \infty we have that Bord (∞,∞)Bord_{(\infty,\infty)} is the symmetric monoidal ∞-groupoid (≃\simeq infinite loop space) Ω ∞MO\Omega^\infty M O that underlies the Thom spectrum.

Its homotopy groups are the cobordism rings

π nBord (∞,∞)≃Ω n. \pi_n Bord_{(\infty,\infty)} \simeq \Omega_n \,.

Therefore a symmetric monoidal ∞\infty-functor

Bord (∞,∞)→S Bord_{(\infty,\infty)} \to S

to some symmetric monoidal ∞\infty-groupoid SS is a genus.

References

General

A specific realization of this idea in terms of (∞,n)-category modeled as n-fold complete Segal space is in (definition 2.2.9, page 36)

In that article a proof of the cobordism hypothesis is indicated. A review is in

Other discussions of higher categories of cobordisms are

In dimension 2

A detailed construction of the (2,2)-category of 2-dimensional cobordisms is

Discussion of a 2-category of complex cobordisms, aimed at formalizing chiral 2d CFT:

In dimension 3

In dimension ∞\infty

For a discussion of the relation of Bord (∞,∞)Bord_{(\infty,\infty)} to the Thom spectrum and the cobordism ring see also

Last revised on December 19, 2021 at 14:09:44. See the history of this page for a list of all contributions to it.