In mathematics, cobordism is a relation between manifolds, based on the idea of boundary. We can say that two manifolds M and N are cobordant if their union is the complete boundary of a third manifold L; L is then called a cobordism between M and N. In this way we get an equivalence relation on manifolds. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... Look up Boundary in Wiktionary, the free dictionary. ... In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ...

For example, if M consists of a circle, and N of two circles, M and N together make up the boundary of a pair of pants L (see the figure at right). Thus the pair of pants is a cobordism between M and N. Circle illustration This article is about the shape and mathematical concept of circle. ... A pair of pants Six pairs of pants sewn together to form an open surface of genus two In mathematics, a pair of pants is a simple two-dimensional surface resembling a pair of pants. ...

A cobordism between a single circle (at the top) and a pair of disjoint circles (at the bottom).

An n-manifold M is said to be null-cobordant if there is a cobordism between M and the empty manifold; in other words, M is the entire boundary of some (n+1)-manifold. For example, the circle is null-cobordant since it bounds a disk. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

The general bordism problem is to calculate the cobordism classes of suitable, more precisely formulated cobordism relations. We should, for example, mention the orientation question: assume all manifolds are smooth and oriented. Then the correct definition is in terms of M and (reversed orientation) making up the boundary of L, with the induced orientations. This article discusses orientability and orientation on surfaces and manifolds. ...

History

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. Lev Semenovich Pontryagin (Russian: Лев Семёнович Понтрягин) (3 September 1908- 3 May 1988) was a Soviet/Russian mathematician. ... René Thom (September 2, 1923 - October 25, 2002) was a French mathematician and founder of the catastrophe theory. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... In mathematics, the Thom space or Thom complex of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space. ... In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ... In mathematics, K-theory is, firstly, an extraordinary cohomology theory which consists of topological K-theory. ... In mathematics, the Hirzebruch-Riemann-Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruchs 1954 result contributing to the Riemann-Roch problem for complex algebraic varieties of all dimensions. ... In the mathematics of manifolds and differential operators, the Atiyah-Singer index theorem is an important unifying result that connects topology and analysis. ...

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. A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. ...

See also

• h-cobordism
• List of cohomology theories
• Symplectic filling

In mathematics, cobordism is a relation between manifolds, based on the idea of boundary. ... This is a list of some of the ordinary and generalized (or extraordinary) homology and cohomology theories in algebraic topology that are defined on the categories of CW complexes or spectra. ... There are very few or no other articles that link to this one. ...

References

• J.F. Adams, Stable homotopy and generalised homology, Univ. Chicago Press (1974)
• D. Quillen, On the formal group laws of unoriented and complex cobordism theory Bull. Amer. Math. Soc, 75 (1969) pp. 1293–1298
• D.C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Acad. Press (1986)
• Yu.B. Rudyak, Cobordism, SpringerLink Encyclopaedia of Mathematics (2001)
• R.E. Stong, Notes on cobordism theory, Princeton Univ. Press (1968)