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.
Cobordism induces an equivalence relation on manifolds: we say that M and N are cobordant if there is a cobordism , and we call W a cobordism between M and N, or from M to N. A basic question is to determine the equivalence classes for this relationship, called the cobordism classes of manifolds. These form a ring called the cobordism ring, and denoted .
The term "bordism" comes from French "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-.
Cobordisms have a natural addition given by disjoint union, and multiplication given by product. They do not have an additive inverse, but cobordism classes do, given by reversing orientation. Thus cobordism classes form a ring, denoted (for cobordisms with a G-structure); note that as disjoint union is cobordant to connect sum, one can also interpret the addition as connect sum.
When there is addition structure, the notion of cobordism must be formulated more precisely: a G-structure on W restricts to a G-structure on M and N, such as for orientation. As a further example, to define complex cobordism one must use stably complex manifolds.
The general definition given here is for (smooth) unoriented cobordism, noted , which is a weak theory in the sense that it is completely determined by ordinary cohomology, specifically, characteristic numbers. Oriented cobordism is similarly a weak theory, but complex cobordism is a strong theory, and does not reduce to ordinary cohomology.
Assume all manifolds are smooth and oriented. Then an oriented cobordism is a manifold W whose boundary (with the induced orientations) is : N has the reversed orientation. This is noted .
Why the reversed orientation? Firstly, because as oriented manifolds, the boundary of the cylinder is : both ends have opposite orientations (this is most familiar for the interval: ). It is also the correct definition in the sense of extraordinary cohomology theory. Lastly, pragmatically, unoriented cobordism is 2-torsion, as (as unoriented manifolds), so reversing the orientation on N yields an integral cobordism theory, instead of a 2-torsion one.
Oriented cobordism is also a weak cobordism theory in the sense of being completely determined by ordinary cohomology, specifically, characteristic numbers.
Null-cobordisms with additional structure are called fillings.
Given a cobordism there exists a smooth function such that . By general position, one can assume is Morse and such that all critical points occur in the interior of . In this setting would be called a Morse function on a cobordism.
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.
Cobordism theories are represented by Thom spectra : given a group G, the Thom spectrum is the Thom space of the tautological bundle over the classifying space : the Thom space of . Note that even for similar groups, Thom spectra can be very different: and are very different, reflecting the difference between oriented and unoriented cobordism.
Unoriented and oriented cobordism are simple theories in the sense that they reduce to ordinary cohomology, as reflected by being determined by characteristic numbers.
From the point of view of spectra, unoriented and oriented cobordism are products of Eilenberg-MacLane spectra:
In low dimensions, the bordism question is trivial, but the category of cobordism is still interesting. For instance, the disk bounding the circle corresponds to a null-ary operation, while the cylinder corresponds to a 1-ary operation and the pair of pants to a binary operation.
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.