nLab cobordism category

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 $M$ to its boundary $\partial M$ 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 $D = Diff_c$ has finite coproducts and the boundary operator $\partial:D\to D$, $M\mapsto \partial M$ 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 $D$ is not actually an additive category). The inclusions $i_M:\partial M\to M$ form a natural transformation of functors $i:\partial\to Id$. Finally, the isomorphism classes of objects in $D$ form a set, so $D$ is essentially small (svelte).

Definition

Axiomatization

Definition

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

• $D$ is a svelte category (i.e. an essentially small category)

  • with finite coproducts (called direct sums, often denoted by $+$),

  • including an initial object $0$ (also often denoted by $\emptyset$),

• $\partial:D\to D$ is an additive (direct-sum-preserving) functor

• and $i:\partial\to Id_D$ is a natural transformation such that $\partial\partial M = 0$ for all objects $M\in D$.

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

Definition

Two objects $M$ and $N$ in a cobordism category $(D,\partial,i)$ are said to be cobordant, written $M\sim_{cob} N$, if there are objects $U,V\in D$ such that $M+\partial U \cong N+\partial V$ where $\cong$ denotes the relation of being isomorphic in $D$.

Remark

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

Definition

Objects of the form $\partial M$ where $M$ is an object in $D$ are said to be boundaries and the objects $V$ such that $\partial V = 0$ 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 $M$ is a boundary and $M\cong N$ then $N$ 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

$$\Omega(D,\partial,i) \,,$$

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

Properties

The homotopy type of the cobordism category

Topological case

Theorem

There is a weak homotopy equivalence

$$\Omega |Cob_d| \simeq \Omega^\infty(MTSO(d))$$

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

$$\Omega^\infty MTSO(d) := {\lim_\to}_{n \to \infty} \Omega^{n+d} Th(U_{d,n}^\perp)$$

where

$$U_{d,n}^\perp = \{ ... \}$$

This is (Galatius-Tillmann-Madsen-Weiss 06, main theorem).

Remark

This statement may be thought of as a limiting case, of the cobordism hypothesis-theorem. See there for more.

Geometric case

• (Ayala)

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

Related concepts

• cobordism, extended cobordism

• category of cobordisms

  • cobordism ring

  • Thom spectrum

  • (∞,n)-category of cobordisms

• conformal cobordism category

• cobordism hypothesis

• cobordism cohomology theory

References

General

A classical reference is

• Robert Stong, Notes on cobordism theory, Princeton University Press 1968 (Russian transl., Mir 1973) (toc pdf, publisher page)

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

• Søren Galatius, Ib Madsen, Ulrike Tillmann, and Michael Weiss, The homotopy type of the cobordism category, Acta Math. 202 (2009), no. 2, 195–239 (arXiv:math/0605249)

A generalization to geometric structure on the cobordisms is discussed in

• David Ayala, Geometric cobordism categories thesis (2009) (arXiv:0811.2280)

Embedded cobordism category

• Oscar Randal-Williams, Embedded Cobordism Categories and Spaces of Manifolds, Int. Math. Res. Not. IMRN 2011, no. 3, 572-608 (arXiv:0912.2505)

On the homotopy groups of the embedded cobordism category:

• Marcel Bökstedt, Anne Marie Svane, A geometric interpretation of the homotopy groups of the cobordism category, Algebr. Geom. Topol. 14 (2014) 1649-1676 (arXiv:1208.3370)

• Marcel Bökstedt, Johan Dupont, Anne Marie Svane, Cobordism obstructions to independent vector fields, Q. J. Math. 66 (2015), no. 1, 13-61 (arXiv:1208.3542)