COBORDISM

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.
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''.

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.
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.

Contents

History

See also

References

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.
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.

See also

★ h-cobordism

★ List of cohomology theories

★ Symplectic filling

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)

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★ R.E. Stong, ''Notes on cobordism theory'', Princeton Univ. Press (1968)