manifolds and cobordisms
cobordism theory, Introduction
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).
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.
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 $D$.
In particular, isomorphic objects are cobordant. Being cobordant is an equivalence relation and for any object $M$ in $D$, one has $\partial M\sim_{cob} 0$.
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.
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.
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
of the cobordism category $(D,\partial,i)$.
There is a weak homotopy equivalence
between the loop space of the geometric realization of the $d$-cobordism category and the Thom spectrum-kind spectrum
where
This is (Galatius-Tillmann-Madsen-Weiss 06, main theorem).
This statement may be thought of as a limiting case, of the cobordism hypothesis-theorem. See there for more.
The Thom group? $\mathcal{N}_*$ of cobordism classes of unoriented compact smooth manifolds is the cobordism semigroup for $D=Diff_c$.
category of cobordisms
A classical reference is
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
Last revised on July 17, 2018 at 03:11:48. See the history of this page for a list of all contributions to it.