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 (French bord, giving cobordism) of a manifold.

Cobordism is a much coarser equivalence relation than diffeomorphism or homeomorphism of manifolds, and is significantly easier to study and compute.

is allowed to have a neighborhood that is homeomorphic to an open subset of the half-space Those points without a neighborhood homeomorphic to an open subset of Euclidean space are the boundary points of

An n-manifold M is called null-cobordant if there is a cobordism between M and the empty manifold; in other words, if M is the entire boundary of some (n + 1)-manifold.

The general bordism problem is to calculate the cobordism classes of manifolds subject to various conditions.

[citation needed] The term bordism comes from French bord, meaning boundary.

Further, cobordism groups form an extraordinary cohomology theory, hence the co-.

In many situations, the manifolds in question are oriented, or carry some other additional structure referred to as G-structure.

, with grading by dimension, addition by disjoint union and multiplication by cartesian product.

Given a cobordism (W; M, N) there exists a smooth function f : W → [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 is called a Morse function on a cobordism.

The cobordism (W; M, N) is a union of the traces of a sequence of surgeries on M, one for each critical point of f. The manifold W is obtained from M × [0, 1] by attaching one handle for each critical point of f. 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 had its roots in the (failed) attempt by Henri Poincaré in 1895 to define homology purely in terms of manifolds (Dieudonné 1989, p. 289).

See Cobordism as an extraordinary cohomology theory for the relationship between bordism and homology.

Bordism was explicitly introduced by Lev Pontryagin in geometric work on manifolds.

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.

In low dimensions, the bordism question is relatively trivial, but the category of cobordism is not.

The set of cobordism classes of closed unoriented n-dimensional manifolds is usually denoted by

of a closed unoriented n-dimensional manifold M is determined by the Stiefel–Whitney characteristic numbers of M, which depend on the stable isomorphism class of the tangent bundle.

[4] Very briefly, the normal bundle ν of an immersion of M into a sufficiently high-dimensional Euclidean space

gives rise to a map from M to the Grassmannian, which in turn is a subspace of the classifying space of the orthogonal group: ν: M → Gr(n, n + k) → BO(k).

The resulting cobordism groups are then defined analogously to the unoriented case.

Unlike in the unoriented cobordism group, where every element is two-torsion, 2M is not in general an oriented boundary, that is, 2[M] ≠ 0 when considered in

It is an oriented cobordism invariant, which is expressed in terms of the Pontrjagin numbers by the Hirzebruch signature theorem.

is the group of bordism classes of pairs (M, f) with M a closed n-dimensional manifold M (with G-structure) and f : M → X a map.

Such pairs (M, f), (N, g) are bordant if there exists a G-cobordism (W; M, N) with a map h : W → X, which restricts to f on M, and to g on N. An n-dimensional manifold M has a fundamental homology class [M] ∈ Hn(M) (with coefficients in

in the oriented case), defining a natural transformation which is far from being an isomorphism in general.

The computation is only easy if the particular cobordism theory reduces to a product of ordinary homology theories, in which case the bordism groups are the ordinary homology groups This is true for unoriented cobordism.

The last-named theory in particular is much used by algebraic topologists as a computational tool (e.g., for the homotopy groups of spheres).