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 are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes from the French word bord for boundary. The boundary of an (n + 1)-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 for smooth (i.e., differentiable) manifolds, but there are now also versions for piecewise-linear and topological manifolds.

A cobordism is a manifold with boundary whose boundary is partitioned in two, .

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 – 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.

Read more about Cobordism:  Surgery Construction, Morse Functions, History, Categorical Aspects, Unoriented Cobordism, Cobordism of Manifolds With Additional Structure, Cobordism As An Extraordinary Cohomology Theory

Other articles related to "cobordism":

Surgery Exact Sequence - Classification of Manifolds
... First there must be a normal cobordism between the degree one normal maps induced by and, this means in ... Denote the normal cobordism ... If the surgery obstruction in to make this normal cobordism to an h-cobordism (or s-cobordism) relative to the boundary vanishes then and in fact represent the same element in the surgery structure set ...
List Of Cohomology Theories - Bordism and Cobordism Theories - Unoriented Cobordism
... Coefficient ring π*(MO) is the ring of cobordism classes of unoriented manifolds, and is a polynomial ring over the field with 2 elements on generators of degree i for every i not of the form 2n−1 ... MO is a rather weak cobordism theory, as the spectrum MO is isomorphic to H(π*(MO)) ("homology with coefficients in π*(MO)") – MO is a product of ... This was the first cobordism theory to be described completely ...
Cobordism As An Extraordinary Cohomology Theory
... Similarly, every cobordism theory has an extraordinary cohomology theory, with homology ("bordism") groups and cohomology ("cobordism") groups for any ... The cobordism groups defined above are, from this point of view, the homology groups of a point ... Such pairs, are bordant if there exists a G-cobordism with a map, which restricts to on, and to on ...