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 (French bord, giving cobordism) 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 an (n + 1)-dimensional manifold W is an n-dimensional manifold ∂W 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 (i.e., differentiable), but there are now also versions for piecewise linear and topological manifolds.
A cobordism between manifolds M and N is a compact manifold W whose boundary is the disjoint union of M and N, .
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 h-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 n-dimensional manifold M is a topological space locally (i.e., near each point) homeomorphic to an open subset of Euclidean space A manifold with boundary is similar, except that a point of M 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 . M and N are called cobordant if such a cobordism exists. All manifolds cobordant to a fixed given manifold M form the cobordism class of M.
Every closed manifold M is the boundary of the non-compact manifold M × [0, 1); for this reason we require W to be compact in the definition of cobordism. Note however that W is not required to be connected; as a consequence, if M = ∂W1 and N = ∂W2, then M and N are cobordant.
Examples
The simplest example of a cobordism is the unit interval . It is a 1-dimensional cobordism between the 0-dimensional manifolds { 0}, { 1}. More generally, for any closed manifold M, (; , ) is a cobordism from M × { 0} to M × { 1}.
If M consists of a circle, and N of two circles, M and N together make up the boundary of a pair of pants W (see the figure at right). Thus the pair of pants is a cobordism between M and N. A simpler cobordism between M and N is given by the disjoint union of three disks.
The pair of pants is an example of a more general cobordism: for any two n-dimensional manifolds M, M′, 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 (n + 1)-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 (n + 1)-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 (compact) 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.