# Bordism

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## Contents |

## [edit] 1 Introduction

The theory of bordism is one of the deepest and most influential parts of
algebraic topology. The foundations of bordism were laid in the pioneering works of Pontrjagin [Pontryagin1959] and Thom [Thom1954], and the theory experienced a spectacular development in the 1960s. In particular, Atiyah [Atiyah1961] showed that bordism is a generalised homology theory and related it to the emergent *K*-theory. The main introductory reference is the monograph [Stong1968].

Basic geometric constructions of bordism and cobordism, as well as homotopical definitions are summarised here. For more information, see the pages in the category **Bordism**.

## [edit] 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 a exists, and
are called *bordant*. The bordism relation splits manifolds
into equivalence classes (see the Figure), which are called
*bordism classes*.

## [edit] 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 . Hence, and 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 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 *-dimensional unoriented bordism group of * and denoted (other notations: , ).

The assignment defines a generalised homology theory, that is, it is functorial in , homotopy invariant, 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 vector -plane bundle . The
*cobordism groups* are defined dually:

where denotes the set of based 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 when one considers generalised homology and cohomology theories.

## [edit] 4 Oriented and complex bordism

The bordism relation may be extended to manifolds endowed with some additional structure, which leads to the most important examples of bordism theories. The universal homotopical framework for geometric bordism with additional structure is provided by the theory of B-bordism.

The simplest additional structure is an orientation. By definition, two oriented -dimensional manifolds and are *oriented bordant* if there is an oriented -dimensional manifold with boundary such that , where denotes with the orientation reversed. The *oriented bordism groups*
and the *oriented bordism ring* are defined
accordingly. Given an oriented manifold , the manifold has a canonical orientation such that . Hence, in . Unlike , elements of
generally do not have order 2.

Complex structure gives another important example of an additional
structure on manifolds. However, a direct attempt to define the
bordism relation on complex manifolds fails because the manifold
is odd-dimensional and therefore cannot be complex. This can
be remedied by considering *stably complex* (also known as
*weakly almost complex*, *stably almost complex* or *quasicomplex*) structures.

Let denote the tangent bundle of , and the product vector bundle
over . A *tangential stably complex structure* on is
determined by a choice of an isomorphism

between the "stable" tangent bundle and a complex vector
bundle over . Some of the choices of such isomorphisms
are deemed to be equivalent, i.e. determine the same stably
complex structures (see details in Chapters II and VII of [Stong1968]). In
particular, two stably complex structures are equivalent if they
differ by a trivial complex summand. A *normal stably complex structure* on is determined by a choice of a complex bundle
structure on the normal bundle of an embedding
. Tangential and normal stably
complex structures on determine each other by means of the
canonical isomorphism . We therefore may restrict
our attention to tangential structures only.

A *stably complex manifold* is a pair consisting of a manifold and a stably complex structure
on it. This is a generalisation of a complex and
*almost complex* manifold (where the latter means a manifold
with a choice of a complex structure on , i.e. a
stably complex structure with ).

**Example 4.1.**
Let . The standard complex structure on is
equivalent to the stably complex structure determined by the
isomorphism

where is the Hopf line bundle. On the other hand, the isomorphism

determines a trivial stably complex structure on .

The bordism relation can be defined between stably complex
manifolds. Like the case of unoriented bordism, the set of
bordism classes of -dimensional stably complex manifolds is an Abelian group with respect to the disjoint union. This group is called the *-dimensional complex bordism group* and
denoted by . The zero is represented by the bordism
class of any manifold which bounds and whose stable tangent
bundle is trivial (and therefore isomorphic to a product complex
vector bundle ). The sphere provides an example
of such a manifold. The opposite element to the bordism class
in the group may be
represented by the same manifold with the stably complex
structure determined by the isomorphism

where is given by .

An abbreviated notation for the complex bordism class will be used whenever the stably complex structure is clear from the context.

The *complex bordism and cobordism groups* of a space
are defined similarly to the unoriented case:

where is the Thom space of the universal complex -plane bundle . These groups are -modules and give rise to a multiplicative (co)homology theory. In particular, is a graded ring.

The graded ring with is
called the *complex cobordism ring*; it has nontrivial
elements only in nonpositively graded components.

## [edit] 5 Connected sum and bordism

For manifolds of positive dimension the disjoint union representing the sum of bordism classes may be replaced by their "connected sum", which represents the same bordism class.

The connected sum of manifolds and of the same dimension is constructed as follows. Choose points and , and take closed -balls and around them (both manifolds may be assumed to be endowed with a Riemannian metric). Fix an isometric embedding of a pair of standard -balls (here ) into which maps onto and onto . If both and are oriented we additionally require the embedding to preserve the orientation on the first ball and reverse in on the second. Now, using this embedding, replace in the pair of balls by a "pipe" . After smoothing the angles in the standard way we obtain a smooth manifold .

If both and are connected the smooth structure on does not depend on a choice of points , and embedding . It does however depend on the orientations; and are not diffeomorphic in general.

There are smooth contraction maps and . In the oriented case the manifold can be oriented in such a way that both contraction maps preserve the orientations.

A bordism between and may be constructed as follows. Consider a cylinder , from which we remove an -neighbourhood of the point . Similarly, remove the neighbourhood from (each of these two neighbourhoods can be identified with the half of a standard open -ball). Now connect the two remainders of cylinders by a "half pipe" in such a way that the half-sphere is identified with the half-sphere on the boundary of , and is identified with the half-sphere on the boundary of . Smoothening the angles we obtain a manifold with boundary (or in the oriented case), see the Figure.

If and are stably complex manifolds, then there is a canonical stably complex structure on , which is constructed as follows. Assume the stably complex structures on and are determined by isomorphisms

Using the isomorphism , we define a stably complex structure on by the isomorphism

This stably complex structure is called the
*connected sum of stably complex structures* on and
. The corresponding complex bordism class is .

## [edit] 6 Structure results

The theory of unoriented (co)bordism was first to be completed: its coefficient ring was calculated by Thom, and the bordism groups of cell complexes were reduced to homology groups of with coefficients in . The corresponding results are summarised as follows:

**Theorem 6.1.**

- Two manifolds are unorientedly bordant if and only if they have identical sets of Stiefel-Whitney characteristic numbers.
- is a polynomial ring over with one generator in every positive dimension .
- For every cell complex the module is a free graded -module isomorphic to .

Parts 1 and 2 were proved in [Thom1954]. Part 3 was proved in [Conner&Floyd1964].

Calculating the complex bordism ring turned out to be a much more difficult problem:

**Theorem 6.2.**

- is a polynomial ring over generated by the bordism classes of complex projective spaces , .
- Two stably complex manifolds are bordant if and only if they have identical sets of Chern characteristic numbers.
- is a polynomial ring over with one generator in every even dimension , where .

Part 1 can be proved by the methods of Thom. Part 2 follows from the results of [Milnor1960] and [Novikov1960]. Part 3 is the most difficult one; it was done in 1960 in [Novikov1960] (see also [Novikov1962] for a more detailed account) and Milnor (unpublished, but see [Thom1995]).

Note that part 3 of Theorem 6.1 does not extend to complex bordism; is not a free -module in general. Unlike the case of unoriented bordism, the calculation of complex bordism of a space does not reduce to calculating the coefficient ring and homology groups .

The calculation of the oriented bordism ring was completed by [Novikov1960] (ring structure modulo torsion) and [Wall1960] (additive torsion), with important contributions made by Rokhlin, Averbuch, and Milnor. Unlike complex bordism, the ring has additive torsion. We give only a partial result here, which does not fully describe the torsion elements. For the complete description of the ring see the Oriented bordism page.

**Theorem 6.3.**

- is a polynomial ring over generated by the bordism classes of complex projective spaces , .
- The subring of torsion elements contains only elements of order 2. The quotient is a polynomial ring over with one generator in every dimension where .
- Two oriented manifolds are bordant if and only if they have identical sets of Pontrjagin and Stiefel-Whitney characteristic numbers.

For more specific information about the three bordism theories, including constructions of manifolds representing polynomial generators in the bordism rings and applications, see the Unoriented bordism, Oriented bordism, and Complex bordism pages.

## [edit] 7 References

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*Differentiable periodic maps*, Academic Press Inc., Publishers, New York, 1964. MR0176478 (31 #750) Zbl 0417.57019 - [Milnor1960] J. Milnor,
*On the cobordism ring and a complex analogue. I*, Amer. J. Math.**82**(1960), 505–521. MR0119209 (22 #9975) Zbl 0095.16702 - [Novikov1960] S. P. Novikov,
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*Determination of the cobordism ring*, Ann. of Math. (2)**72**(1960), 292–311. MR0120654 (22 #11403) Zbl 0097.38801