Cobordisms

Cobordism

Suppose and are two closed manifolds of dimension , then a cobordism between and is a compact -manifold whose boundary is . If such exists, we say and are cobordant to each other.

Oriented Cobordism

Let and be two oriented closed manifolds of dimension . An oriented cobordism from to is a compact oriented manifold together with orientation-preserving smooth maps such that maps diffeomorphically onto the in/out-boundary of .

Cobordism Equivalence

Two oriented cobordisms and from to are equivalent if there is an orientation-preserving diffeomorphism such that the following diagram commutes: Clearly this is an equivalent relation and we can accordingly define a cobordism class.

Axiomatic TQFTs

Atiyah first gave a set of axioms for a topological quantum field theory in 1988, for which is now characterized as a symmetric monoidal functor. The following is a modernized version of Atiyah’s original axioms:

An -dimensional topological quantum field theory (TQFT) is a rule that assigns to each closed -dimensional manifold a vector space , and to each -dimensional cobordism a linear map , such that the following five axioms hold:

1. Two equivalent cobordisms must have the same image;
2. The cylinder must be sent to the identity map on ;
3. preserves the decomposition of cobordisms;
4. Disjoint union of manifolds must be sent to the tensor product of vector spaces, and disjoint union of cobordisms must be sent to the tensor product of linear maps;
5. The empty manifold must be sent to the ground field .

The first two axioms guarantee that the theory is “topological”, in the sense that it does not depend on any additional structure like metric or curvature. The fourth axiom is a standard principle of quantum mechanics that the state space of two independent systems is the tensor product of the two state spaces.

Proposition

The state space assigned to a closed -manifold is always finite-dimensional.

Proof To show is finite dimensional, it suffices to find a spanning set for . Note that we have the following diagram commutes: is a copairing , suppose that is the finite sum , then for each , there holds This means forms a spanning set for .

The Cobordism Category

Cobordism Category

The cobordism category is the category whose objects are closed -manifolds, and morphisms are cobordism classes between them. The composition of morphisms is given by gluing of cobordisms, and the identity morphism on is given by the cobordism class represented by the cylinder .

Cobordism Classes Induced by Diffeomorphisms

Proposition

Two diffeomorphisms induce the same cobordism class if and only if they are smoothly homotopic.

Proof We have the following diagram commute as , are homotopic: Let us define so that the diagram commute. This shows that the induced cobordisms are equivalent. The other implication follows by composing with the projection , we get a homotopy from to .

Disjoint Union and Twists

Twist Map

Functorial TQFTs

Topological Quantum Field

An -dimensional topological quantum field is a symmetric monoidal functor from to .