Manifolds and cobordisms

Monoidal categories



An (oriented) cobordism Σ\Sigma from an (oriented) smooth manifold X inX_{in} to an (oriented) smooth manifold X outX_{out} is a smooth manifold with boundary Σ\Sigma such that its boundary is the disjoint union

ΣX inX out \partial \Sigma \simeq X_{in} \coprod X_{out}

with induced orientation agreeing with the given one on X inX_{in} and being the opposite of that of X outX_{out}. Hence by labelling disjoint components of the boundary of any manifold with boundary Σ\Sigma as either “incoming” or “outgoing”, then Σ\Sigma becomes a cobordism from its incoming to its outgoing boundary components.

(While X inX outX_{in} \coprod X_{out} is the boundary of Σ\Sigma, conversely Σ\Sigma is the “co-boundary” of X inX outX_{in}\coprod X_{out}. This is part of the reason for the “co-” in “cobordism”, but sometimes one just says bordism. The difference is more pronounced when distinguishing between bordism homology theory and cobordism cohomology theory.)

With a little bit of technical conditions on the boundary inclusions added, then (n1)(n-1)-dimensional manifolds with diffeomorphism classes of nn-dimensional cobordisms between them form a category (a category of cobordisms) whose objects are the manifolds, whose morphisms are the cobordisms, and whose composition operation is the operation of gluing two cobordisms along a common boundary component. Such a category of cobordisms Bord nBord_n of some dimension nn is naturally a symmetric monoidal category Bord n Bord_n^{\sqcup} with the tensor product being the disjoint union \sqcup of manifolds.

The connected components in this category are called cobordism classes of manifolds. Under cartesian product and disjoint union, these form what is called the cobordism ring in the given dimension. The study of these classes, hence of manifolds “up to cobordisms”, is a central topic in algebraic topology.

A central insight connecting algebraic topology with mathematical physics is that a strong monoidal functor on a category of cobordisms with values in something like the category Vect (with its standard tensor product of modules) may be thought of as a formalized incarnation of what in physics is called a topological quantum field theory

Z:Bord n Vect . Z \colon Bord_n^\sqcup \longrightarrow Vect^\otimes \,.

Here one thinks of a cobordism Σ\Sigma as a piece of spacetime (or worldvolume) of dimension nn, and of the (n1)(n-1)-dimensional manifolds that this goes between as a piece of space (or brane). A functor ZZ as above is then thought of as sending each (n1)(n-1)-dimensional space XX to its space of quantum states Z(X)Z(X) and each spacetime Σ\Sigma between X inX_{in} and X outX_{out} to a linear map Z(Σ):Z(X in)Z(X out)Z(\Sigma)\colon Z(X_{in}) \longrightarrow Z(X_{out}).

From the perspective of mathematics, such a functor is a way to break up diffeomorphism invariants of closed manifolds of dimension nn into pieces and being able to reconstruct them from gluing of data associated to manifolds with (n1)(n-1)-dimensional boundary.

If one considers manifolds with corners then there is a fairly evident refinement of the concept of cobordism that allows to further refine this process to gluing of extended cobordisms of any dimension kk going between extended cobordisms of dimension k1k-1. Such extended cobordisms of maximal dimension nn form a symmetric monoidal (infinity,n)-category called, naturally, the (infinity,n)-category of cobordisms. The cobordism hypothesis asserts that this is a most fundamental object in higher category theory and higher algebra, namely that it is the free symmetric monoidal (infinity,n)-category with duals. The corresponding extended concept of topological quantum field theory is accordingly called extended TQFT or similar.


Revised on June 26, 2017 05:42:17 by Urs Schreiber (