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cobordism category in nLab

Contents

Context

Manifolds and Cobordisms

manifolds and cobordisms

cobordism theory, Introduction

Definitions

Genera and invariants

Classification

Theorems

Functorial Quantum Field Theory

Contents

Idea

The notion of cobordism categories in the original sense of Stong 1968 abstracts basic properties of (variants of) categories whose objects are compact manifolds with boundary, with the intent of regarding these as cobordisms between their boundary components.

A closely related but nominally different notion are categories whose morphisms are taken to be cobordisms between their boundary components.

Beware that the use of terminology not always brings out this distinction; but these days the second meaning is more prevalent, in particular in discussion of cobordism cohomology and of topological field theory.

Definitions

Cobordisms as objects

The axiomatization below is motivated as capturing the following familiar situation:

The category DD of compact smooth manifolds with boundary, has finite coproducts and the boundary operator ∂:D→D\partial \colon D\to D, M↦∂MM\mapsto \partial M is an endofunctor commuting with coproducts. (Often these coproducts are referred to as direct sums. Notice that DD is similar to but not actually an additive category. The inclusions i M:∂M→Mi_M \colon \partial M\to M form a natural transformation of functors i:∂→Idi \colon \partial\to Id. Finally, the isomorphism classes of objects in DD form a set, so DD is essentially small (svelte).

Definition

A Stong cobordism category is a triple (D,∂,i)(D,\partial,i) where

Stong 1968, p. 4.

Definition

Two objects MM and NN in a cobordism category (D,∂,i)(D,\partial,i) are said to be cobordant, written M∼ cobNM\sim_{cob} N, if there are objects U,V∈DU,V\in D such that M+∂U≅N+∂VM+\partial U \cong N+\partial V where ≅\cong denotes the relation of being isomorphic in DD.

Definition

Objects of the form ∂M\partial M where MM is an object in DD are said to be boundaries and the objects VV such that ∂V=0\partial V = 0 are said to be closed.

Definition

By the above, the relation of being cobordant is compatible with the direct sum, in the sense that the direct sum induces an associative commutative operation on the set of equivalence classes, which hence becomes a commutative monoid called the cobordism semigroup

Ω(D,∂,i), \Omega(D,\partial,i) \,,

of the cobordism category (D,∂,i)(D,\partial,i).

Cobordisms as morphisms

(…)

e.g. GMWT09, 2.1

(…)

Properties

The following properties concern the notion ob cobordism categories with cobordisms serving as morphisms.

The homotopy type of the cobordism category

Topological case

Theorem

There is a weak homotopy equivalence

Ω|Cob d|≃Ω ∞(MTSO(d)) \Omega |Cob_d| \simeq \Omega^\infty(MTSO(d))

between the loop space of the geometric realization of the dd-cobordism category and the Thom spectrum-kind spectrum

Ω ∞MTSO(d):=lim → n→∞Ω n+dTh(U d,n ⊥) \Omega^\infty MTSO(d) := {\lim_\to}_{n \to \infty} \Omega^{n+d} Th(U_{d,n}^\perp)

where

U d,n ⊥={...} U_{d,n}^\perp = \{ ... \}

This is Galatius, Tillmann, Madsen & Weiss 2009, main theorem.

Geometric case

The Thom group? 𝒩 *\mathcal{N}_* of cobordism classes of unoriented compact smooth manifolds is the cobordism semigroup for D=Diff cD=Diff_c.

Examples

Examples of cobordism categories besides those of manifolds with ( B , f ) (B,f) -structure:

See also MO:q/59677.

References

General

The notion of cobordism categories with cobordisms as objects is due to

Most authors these days use “cobordism category” to refer to the notion where cobordisms form the morphisms:

The GMTW theorem about the homotopy type of the cobordisms category with topological structures on the cobordisms appears in

A generalization to geometric structure on the cobordisms is discussed in

Embedded cobordism category

On the homotopy groups of the embedded cobordism category:

Last revised on March 28, 2024 at 17:29:38. See the history of this page for a list of all contributions to it.