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We denote the unoriented bordism groups by . The sum of these groups
forms a ring under cartesian products of manifolds. Thom [Thom1954] has shown that this ring is a polynomial ring over in variables for and he has shown that for even one can take for . Dold [Dold1956] has constructed manifolds for with odd.
2 Construction and examples
Dold constructs certain bundles over with fibre denoted by
Using the results by Thom [Thom1954] Dold shows that these manifolds give ring generators of .
Theorem (Dold) [Dold1956] 2.1. For even set and for set . Then for
are polynomial generators of olver :
To prove the Theorem Dold has to compute the characteristic numbers which according to Thom's theorem determine the bordism class. As a first step Dold computes the cohomology ring with -coeffcients. The fibre bundle has a section and we consider the cohomology classes (always with -coefficients)
where is a generator of , and
which is characterized by the property that the restriction to a fibre is non-trivial and .
Theorem [Dold1956] 3.1. The classes and generate with only the relations
The Steenrod squares act by
and all other Squares act trivially on and . On the decomposable classes the action is given by the Cartan formula.
The total Stiefel-Whitney class of the tangent bundle is
To give explicit polynomial generators is useful information, if one wants to prove a formula like for example that , where is the -th Stiefel-Whitney class of an -dimensional manifold and is the Euler characteristic, which one has to check on generators. But it does not help to classify manifolds up to bordism. There is an explicit answer to this question by Thom:
Theorem [Thom1954] 4.1. Two closed -manifolds and are bordant if and only if all Stiefel-Whitney numbers agree:
for all partitions .
5 Further discussion
For odd the manifolds are orientable and thus after choosing an orientation give an element in the oriented bordism group . Since admits an obvious orientation reversing diffeomorphism, these elements are -torsion. Thus we obtain a subring in isomorphic to . For more information about see the page on oriented bordism.
- [Dold1956] A. Dold, Erzeugende der Thomschen Algebra , Math. Z. 65 (1956), 25–35. MR0079269 (18,60c) Zbl 0071.17601
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502