manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
The notion of cobordism categories in the original sense of Stong 1968 abstracts basic properties of (variants of) categories whose objects are compact manifolds with boundary, with the intent of regarding these as cobordisms between their boundary components.
A closely related but nominally different notion are categories whose morphisms are taken to be cobordisms between their boundary components.
Beware that the use of terminology not always brings out this distinction; but these days the second meaning is more prevalent, in particular in discussion of cobordism cohomology and of topological field theory.
The axiomatization below is motivated as capturing the following familiar situation:
The category $D$ of compact smooth manifolds with boundary, has finite coproducts and the boundary operator $\partial \colon D\to D$, $M\mapsto \partial M$ is an endofunctor commuting with coproducts. (Often these coproducts are referred to as direct sums. Notice that $D$ is similar to but not actually an additive category. The inclusions $i_M \colon \partial M\to M$ form a natural transformation of functors $i \colon \partial\to Id$. Finally, the isomorphism classes of objects in $D$ form a set, so $D$ is essentially small (svelte).
A Stong 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 $(D,\partial,i)$.
(…)
e.g. GMWT09, 2.1
(…)
The following properties concern the notion ob cobordism categories with cobordisms serving as morphisms.
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 2009, 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 $D=Diff_c$.
Examples of cobordism categories besides those of manifolds with $(B,f)$-structure:
See also MO:q/59677.
category of cobordisms
The notion of cobordism categories with cobordisms as objects is due to
Most authors these days use “cobordism category” to refer to the notion where cobordisms form the morphisms:
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
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)
Last revised on March 28, 2024 at 17:29:38. See the history of this page for a list of all contributions to it.