nLab
cobordism category

Context

Manifolds and Cobordisms

Functorial Quantum Field Theory

Contents

Idea

The notion of cobordism category is an abstract one intended to capture important features of (many variants of) the category of cobordisms and include in the same formalism cobordisms for closed manifolds with various kinds of structure.

The passage from a manifold MM to its boundary M\partial M has some formal properties which are preserved in the presence of orientation, for manifolds with additional structure and so on. The category of compact smooth manifolds with boundary D=Diff cD = Diff_c has finite coproducts and the boundary operator :DD\partial:D\to D, MMM\mapsto \partial M is an endofunctor commuting with coproducts. (Often these coproducts are referred to as direct sums, and some say that \partial is an additive functor, but DD is not actually an additive category). The inclusions i M:MMi_M:\partial M\to M form a natural transformation of functors i:Idi:\partial\to Id. Finally, the isomorphism classes of objects in DD form a set, so DD is essentially small (svelte).

Definition

Axiomatization

Definition

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

Note that ii is not required to be a subfunctor of the identity, i.e. the components i Mi_M are not required to be monic, which is however often the case in examples.

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,VDU,V\in D such that M+UN+VM+\partial U \cong N+\partial V where \cong denotes the relation of being isomorphic in DD.

Remark

In particular, isomorphic objects are cobordant. Being cobordant is an equivalence relation and for any object MM in DD, one has M cob0\partial M\sim_{cob} 0.

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.

Remark

In particular, every boundary is closed. A direct sum of closed objects (resp. boundaries) is a closed object (resp. a boundary). If an object MM is a boundary and MNM\cong N then NN is also a boundary.

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).

Properties

The homotopy type of the cobordism category

Topological case

Theorem

There is a weak homotopy equivalence

ΩCob dΩ (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 nΩ 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 (GalatiusTillmannMadsenWeiss, 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.

References

A classical reference is

  • Robert E. Stong, Notes on cobordism theory, Princeton University Press 1968 (Russian transl., Mir 1973)

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

Revised on April 25, 2013 15:58:48 by Urs Schreiber (82.169.65.155)