Oriented bordism
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1 Introduction
By the Pontrjagin-Thom isomorphism the oriented bordism groups of closed oriented manifolds are isomorphic to the homotopy groups of the Thom spectrum .
[edit] 2 Generators
- , generated by a point.
- , as circles bound disks.
- , as oriented surfaces bound handlebodies.
- .
- , generated by the complex projective plane , detected by the signature.
- , generated by the Wu manifold , detected by the deRham invariant.
- .
- generated by and .
for : see also [Milnor&Stasheff1974, p. 203].
is a polynomial ring, with generators , detected by the Pontrjagin numbers.
is an integral polynomial ring with generators the ``Milnor hypersurfaces``.
[edit] 3 Invariants
The signature of a closed oriented manifold is a fundamental bordism invariant defining a ring homomorphism
(Note that manifolds of dimension not divisible by 4 have signature zero.)
For a muli-index of degree the Pontryagin number of a closed, oriented manifold of dimension is the integer
where is the k-the Pontrjagin of and its fundamental class. The Stiefel-Whitney numbers of , , are defined similarly using Stiefel-Whitney classes. These numbers are bordism invariants (see for example [Milnor&Stasheff1974, Theorm 4.9, Lemma 17.3]) and clearly additive. Hence we have homomorphisms
By Hirzebruch's signature theorem [Hirzebruch1953], [Hirzebruch1995, Theorem 8.2.2], there is a certain rational polynomial in the Pontrjagin classes, called the L-polynomial , which computes the signature of M
For example:
[edit] 4 Classification
Thom [Thom1954] computed . This is equivalent to the computation of the rational (co)homology of , as shown here. The cohomology is a polynomial ring with generators the Pontryagin classes, so that Pontryagin numbers give an additive isomorphism . Since all products of have linearly independent collections of Pontryagin numbers, there is a ring isomorphism from to a polynomial ring with generators .
Independently Averbuch [Averbuh1959] and Milnor [Milnor1960] showed that has no odd torsion. In addition, Novikov [Novikov1960] showed that is isomorphic to a polynomial ring . Here a generator can be any -dimensional manifold such that the Pontryagin number equals , if is not a prime power, or equals , if is a power of the prime . (Here is the polynomial which expresses in terms of the elementary symmetric polynomials of the .)
Wall [Wall1960], using earlier results of [Rokhlin1953], determined the structure of completely. In particular he proved the following theorems.
Theorem 3.1 [Wall1960, Theorem 2]. All torsion in is of exponent 2.
Theorem 3.2 [Wall1960, Corollary 1]. Two closed oriented n-manifolds and are oriented cobordant if and only if they have the same Stiefel-Whitney and Pontryagin numbers:
For the complete ring structure, we first describe the subalgebra of the unoriented bordism ring consisting of classes which contain a manifold whose first Stiefel-Whitney class is the reduction of an integral class. is a polynomial ring with coefficients on generators where neither nor are powers of 2, together with generators where is a power of 2. These generators can be described explicitly as follows:
- For with integers and (i.e. not a power of 2), the generator is the Dold manifold and the generator is the mapping torus of the map given by the reflection of at the equator.
- For not a power of 2, the generator is the mapping torus of a certain involution . Indeed any Dold manifold has the involution .
- For a power of 2, the generator is . This generator is also represented by .
Now there is an exact sequence
where the ring homomorphism is induced by the forgetful map , and is the derivation .
Together with the result that one can choose generators for such that , this determines the ring structure of .
[edit] 5 References
- [Averbuh1959] B. G. Averbuh, Algebraic structure of cobordism groups, Dokl. Akad. Nauk SSSR 125 (1959), 11–14. MR0124894 (23 #A2204)
- [Hirzebruch1953] F. Hirzebruch, Über die quaternionalen projektiven Räume, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1953 (1953), 301–312 (1954). MR0065155 (16,389a) Zbl 0057.15503
- [Hirzebruch1995] F. Hirzebruch, Topological methods in algebraic geometry, Springer-Verlag, Berlin, 1995. MR1335917 (96c:57002) Zbl 0843.14009
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [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, Some problems in the topology of manifolds connected with the theory of Thom spaces, Soviet Math. Dokl. 1 (1960), 717–720. MR0121815 (22 #12545) Zbl 0094.35902
- [Rokhlin1953] V. A. Rohlin, Intrinsic homologies, Doklady Akad. Nauk SSSR (N.S.) 89 (1953), 789–792. MR0056292 (15,53b)
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502
- [Wall1960] C. T. C. Wall, Determination of the cobordism ring, Ann. of Math. (2) 72 (1960), 292–311. MR0120654 (22 #11403) Zbl 0097.38801