Bordism
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Contents |
1 Introduction
The theory of bordism is is one of the most deep and influential parts of the algebraic topology, which experienced a spectacular development in the 1960s. The main introductory reference is the monograph [Stong1968].
Basic geometric constructions of bordisms and cobordisms, as well as homotopical definitions are summarised here. For the more specific information, see B-Bordism and pages on specific bordism theories, such as unoriented, oriented and complex.
2 The bordism relation
All manifolds here are assumed to be smooth, compact and closed (without boundary), unless otherwise specified. Given two -dimensional manifolds and , a bordism between them is an -dimensional manifold with boundary, whose boundary is the disjoint union of and , that is, . If such exists, and are called bordant. The bordism relation splits manifolds into equivalence classes (see Figure), which are called bordism classes.
3 Unoriented bordism
We denote the bordism class of by , and denote by the set of bordism classes of -dimensional manifolds. Then is an abelian group with respect to the disjoint union operation: . Zero is represented by the bordism class of an empty set (which is counted as a manifold in any dimension), or by the bordism class of any manifold which bounds. We also have , so that is a 2-torsion group.
Set . The product of bordism classes, namely , makes a graded commutative ring known as the unoriented bordism ring.
For any (good) space the bordism relation can be extended to maps of -dimensional manifolds to : two maps and are bordant if there is a bordism between and and the map extends to a map . The set of bordism classes of maps forms an abelian group called the group of -dimensional unoriented bordisms of and denoted (other notations: , ).
The assignment defines a generalised homology theory, that is, satisfies the homotopy invariance, has the excision property and exact sequences of pairs. For this theory we have , and is an -module.
The Pontrjagin--Thom construction reduces the calculation of the bordism groups to a homotopical problem:
where , and is the Thom space of the universal -dimensional bundle . The cobordism groups are defined dually: \[
O^n(X)=\lim_{k\to\infty}[\Sigma^{k-n}(X_+),MO(k)]
\] where denotes the set of homotopy classes of maps from to . The resulting generalised cohomology theory is multiplicative, which implies that is a graded commutative ring. It follows from the definitions that . The graded ring with is called the unoriented cobordism ring. It has nonzero elements only in nonpositively graded components. The bordism ring and the cobordism ring differ only by their gradings, so the notions of the ``bordism class and ``cobordism class of a manifold are interchangeable. The difference between bordism and cobordism appears only for nontrivial spaces .
3.1 References
- [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010
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