Cobordism

Cobordism

In mathematics, an n+1 cobordism is a triple (W,M,N), where W is an (n+1)-dimensional manifold, whose boundary is the disjoint union of the n-dimensional manifolds M and N. In other words, it is a manifold with boundary whose boundary is partitioned in two.

If the manifolds in question have an additional G-structure (such as an orientation), one can define a "cobordism with G-structure", but there are various technicalities.

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 (W,M,N), 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 Omega^*_G.

Cobordisms are objects of study in their own right, apart from cobordism classes. Categorically, one can think of a cobordism (W,M,N) as a map from M to N, and thus define categories of cobordisms.

Terminology

When one wishes to distinguish the study of cobordism classes from the study of cobordisms as objects in their own right, one calls the equivalence question "bordism of manifolds", and the study of cobordisms as objects "cobordisms of manifolds".

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-.

Examples

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. More generally, this picture shows that for any two manifolds, the disjoint union M coprod N and connected sum M # N are cobordant.

Cobordism classes

The general bordism problem is to calculate the cobordism classes of various cobordism relations.

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 Omega^*_G (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 Omega^*_{text{O}}, 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.

Oriented cobordism

Assume all manifolds are smooth and oriented. Then an oriented cobordism is a manifold W whose boundary (with the induced orientations) is M coprod bar N: N has the reversed orientation. This is noted Omega^*_{mbox{SO}}.

Why the reversed orientation? Firstly, because as oriented manifolds, the boundary of the cylinder M times I is M coprod bar M: both ends have opposite orientations (this is most familiar for the interval: partial I = {1}-{0}). It is also the correct definition in the sense of extraordinary cohomology theory. Lastly, pragmatically, unoriented cobordism is 2-torsion, as 2M = partial(M times I) (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-cobordant

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. Equivalently, if its cobordism class is trivial.

For example:

  • the circle (and more generally, n-sphere) are null-cobordant since they bound an (n+1)-disk
  • every orientable surface is null-cobordant because it is the boundary of a handlebody

Null-cobordisms with additional structure are called fillings.

Connection with Morse and Handlebody theory

Given a cobordism (W,M,N) there exists a smooth function f : W to [0,1] such that f^{-1}(0) = M, f^{-1}(1) = N. By general position, one can assume f is Morse and such that all critical points occur in the interior of W. In this setting f 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 f' give rise to a handle presentation of the triple (W,M,N). 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 as an extraordinary cohomology theory

Together with K-theory, cobordism theory is the basic extraordinary cohomology theory. The cobordism theories defined above are, for this point of view, the cobordism theory of a point: Omega^n_G = Omega^n_G(*). By the other Eilenberg-Steenrod axioms, the cobordism cohomology of a space can be effectively computed once one has computed the cobordism theory of a point. There are also dual cobordism homology theories.

Cobordism theories are represented by Thom spectra MG: given a group G, the Thom spectrum is the Thom space of the tautological bundle EG over the classifying space BG: the Thom space of EG to BG. Note that even for similar groups, Thom spectra can be very different: MSO and MO 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:

MO = H(pi_*(MO)) and MSO = H(pi_*(MSO))

Categories of cobordisms

Cobordisms form a category, indeed a groupoid, where the objects are closed manifolds and the morphisms are cobordisms. Composition is given by gluing together cobordisms end-to-end: the composition of (W,M,N) and (W',N,P) is defined by gluing the right end of the first to the left end of the second, yielding (W'cup_N W,M,P). Inverse is given by switching the labeling of the two ends ("turning the cobordism around"): (W,M,N) mapsto (W,N,M), and the identity elements are given by cylinders (Mtimes I, M, M).

A topological quantum field theory is a functor from a category of cobordisms to a category of vector spaces.

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.

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

Footnotes

References

  • J.F. Adams, Stable homotopy and generalised homology, Univ. Chicago Press (1974)
  • A. Kosinski, Differential Manifolds, Dover Publications (October 19, 2007)
  • 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)
  • R.E. Stong, Notes on cobordism theory, Princeton Univ. Press (1968)

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