Cobordism


In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher.
The boundary of a compact -dimensional manifold is an -dimensional manifold that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds, but there are now also versions for
piecewise linear and topological manifolds.
A cobordism between manifolds and is a compact manifold whose boundary is the disjoint union of and,.
Cobordisms are studied both for the equivalence relation that they generate, and as objects in their own right. Cobordism is a much coarser equivalence relation than diffeomorphism or homeomorphism of manifolds, and is significantly easier to study and compute. It is not possible to classify manifolds up to diffeomorphism or homeomorphism in dimensions ≥ 4 – because the word problem for groups cannot be solved – but it is possible to classify manifolds up to cobordism. Cobordisms are central objects of study in geometric topology and algebraic topology. In geometric topology, cobordisms are [|intimately connected] with Morse theory, and -cobordisms are fundamental in the study of high-dimensional manifolds, namely surgery theory. In algebraic topology, cobordism theories are fundamental extraordinary cohomology theories, and categories of cobordisms are the domains of topological quantum field theories.

Definition

Manifolds

Roughly speaking, an -dimensional manifold is a topological space locally homeomorphic to an open subset of Euclidean space. A manifold with boundary is similar, except that a point of is allowed to have a neighborhood that is homeomorphic to an open subset of the half-space
Those points without a neighborhood homeomorphic to an open subset of Euclidean space are the boundary points of ; the boundary of is denoted by. Finally, a closed manifold is, by definition, a compact manifold without boundary.

Cobordisms

An -dimensional cobordism is a quintuple consisting of an -dimensional compact differentiable manifold with boundary, ; closed -manifolds , ; and embeddings, with disjoint images such that
The terminology is usually abbreviated to. and are called cobordant if such a cobordism exists. All manifolds cobordant to a fixed given manifold form the cobordism class of.
Every closed manifold is the boundary of the non-compact manifold ; for this reason we require to be compact in the definition of cobordism. Note however that is not required to be connected; as a consequence, if and, then and are cobordant.

Examples

The simplest example of a cobordism is the unit interval. It is a 1-dimensional cobordism between the 0-dimensional manifolds,. More generally, for any closed manifold, is a cobordism from to.
If consists of a circle, and of two circles, and together make up the boundary of a pair of pants . Thus the pair of pants is a cobordism between and. A simpler cobordism between and is given by the disjoint union of three disks.
The pair of pants is an example of a more general cobordism: for any two -dimensional manifolds,, the disjoint union is cobordant to the connected sum The previous example is a particular case, since the connected sum is isomorphic to The connected sum is obtained from the disjoint union by surgery on an embedding of in, and the cobordism is the trace of the surgery.

Terminology

An n-manifold M is called null-cobordant if there is a cobordism between M and the empty manifold; in other words, if M is the entire boundary of some -manifold. For example, the circle is null-cobordant since it bounds a disk. More generally, a n-sphere is null-cobordant since it bounds a -disk. Also, every orientable surface is null-cobordant, because it is the boundary of a handlebody. On the other hand, the 2n-dimensional real projective space is a closed manifold that is not the boundary of a manifold, as is explained below.
The general bordism problem is to calculate the cobordism classes of manifolds subject to various conditions.
Null-cobordisms with additional structure are called fillings. Bordism and cobordism are used by some authors interchangeably; others distinguish them. When one wishes to distinguish the study of cobordism classes from the study of cobordisms as objects in their own right, one calls the equivalence question bordism of manifolds, and the study of cobordisms as objects cobordisms of manifolds.
The term bordism comes from French wikt:bord, meaning boundary. Hence bordism is the study of boundaries. Cobordism means "jointly bound", so M and N are cobordant if they jointly bound a manifold; i.e., if their disjoint union is a boundary. Further, cobordism groups form an extraordinary cohomology theory, hence the co-.

Variants

The above is the most basic form of the definition. It is also referred to as unoriented bordism. In many situations, the manifolds in question are oriented, or carry some other additional structure referred to as G-structure. This gives rise to [|"oriented cobordism"] and "cobordism with G-structure", respectively. Under favourable technical conditions these form a graded ring called the cobordism ring, with grading by dimension, addition by disjoint union and multiplication by cartesian product. The cobordism groups are the coefficient groups of a [|generalised homology theory].
When there is additional structure, the notion of cobordism must be formulated more precisely: a G-structure on W restricts to a G-structure on M and N. The basic examples are G = O for unoriented cobordism, G = SO for oriented cobordism, and G = U for complex cobordism using stably complex manifolds. Many more are detailed by Robert E. Stong.
In a similar vein, a standard tool in surgery theory is surgery on normal maps: such a process changes a normal map to another normal map within the same bordism class.
Instead of considering additional structure, it is also possible to take into account various notions of manifold, especially piecewise linear and topological manifolds. This gives rise to bordism groups, which are harder to compute than the differentiable variants.

Surgery construction

Recall that in general, if X, Y are manifolds with boundary, then the boundary of the product manifold is.
Now, given a manifold M of dimension n = p + q and an embedding define the n-manifold
obtained by surgery, via cutting out the interior of and gluing in along their boundary
The trace of the surgery
defines an elementary cobordism. Note that M is obtained from N by surgery on This is called reversing the surgery.
Every cobordism is a union of elementary cobordisms, by the work of Marston Morse, René Thom and John Milnor.

Examples

As per the above definition, a surgery on the circle consists of cutting out a copy of and gluing in The pictures in Fig. 1 show that the result of doing this is either again, or two copies of
For surgery on the 2-sphere, there are more possibilities, since we can start by cutting out either or

Morse functions

Suppose that f is a Morse function on an -dimensional manifold, and suppose that c is a critical value with exactly one critical point in its preimage. If the index of this critical point is p + 1, then the level-set N := f−1 is obtained from M := f−1 by a p-surgery. The inverse image W := f−1 defines a cobordism that can be identified with the trace of this surgery.

Geometry, and the connection with Morse theory and handlebodies

Given a cobordism there exists a smooth function f : W → such that f−1 = M, f−1 = N. By general position, one can assume f is Morse and such that all critical points occur in the interior of W. In this setting f is called a Morse function on a cobordism. The cobordism is a union of the traces of a sequence of surgeries on M, one for each critical point of f. The manifold W is obtained from M × by attaching one handle for each critical point of f.
File:Cobordism.svg|thumb|The 3-dimensional cobordism between the 2-sphere and the 2-torus with N obtained from M by surgery on and W obtained from M × I by attaching a 1-handle
The Morse/Smale theorem states that for a Morse function on a cobordism, the flowlines of give rise to a handle presentation of the triple. Conversely, given a handle decomposition of a cobordism, it comes from a suitable Morse function. In a suitably normalized setting this process gives a correspondence between handle decompositions and Morse functions on a cobordism.

History

Cobordism had its roots in the attempt by Henri Poincaré in 1895 to define homology purely in terms of manifolds. Poincaré simultaneously defined both homology and cobordism, which are not the same, in general. See [|Cobordism as an extraordinary cohomology theory] for the relationship between bordism and homology.
Bordism was explicitly introduced by Lev Pontryagin in geometric work on manifolds. It came to prominence when René Thom showed that cobordism groups could be computed by means of homotopy theory, via the Thom complex construction. Cobordism theory became part of the apparatus of extraordinary cohomology theory, alongside K-theory. It performed an important role, historically speaking, in developments in topology in the 1950s and early 1960s, in particular in the Hirzebruch–Riemann–Roch theorem, and in the first proofs of the Atiyah–Singer index theorem.
In the 1980s the category with compact manifolds as objects and cobordisms between these as morphisms played a basic role in the Atiyah–Segal axioms for topological quantum field theory, which is an important part of quantum topology.