Spin bordism
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1 Introduction
By the Pontrjagin-Thom isomorphism the spin bordism groups of closed manifolds with spin structures are isomorphic to the homotopy groups of the Thom spectrum .
Preliminary results were by Novikov. The main calculation was achieved in [Anderson&Brown&Peterson1966] and [Anderson&Brown&Peterson1967] where it is shown that all torsion is of exponent 2, being of two types: that arising by products with a framed and that which maps monomorphically into unoriented cobordism.
2 Invariants
The spin bordism class of a manifold is detected by -cohomology (Stiefel-Whitney) and KO-theory (Pontryagin) characteristic numbers. (Since a spin structure induces a KO-orientation, we can evaluate polynomials in the KO-Pontryagin classes to get characteristic numbers in .) These characteristic numbers can be defined as the indices of Clifford-linear Dirac operators twisted with the corresponding vector bundles. See [Lawson&Michelsohn1989].
3 Classification
is an equivalence after inverting 2. Thus there is no odd torsion in the spin cobordism groups, and all summands are in degrees divisible by 4.
For where all , we set and . Using the Thom isomorphism we get a map , for which Anderson,Brown and Peterson show that it factorizes through if is even and if is odd. Hence the corresponding characteristic numbers vanish for manifolds of smaller dimension.
Theorem [Anderson&Brown&Peterson1967] 3.1. There are classes such that there is a 2-local homotopy equivalence
From this one can compute the additive structure completely. Concerning the multiplicative structure, is the subring of an integral polynomial ring on classes (dimension 4i) consisting of all classes of dimension a multiple of 8 and twice the classes whose dimension is not a multiple of 8 [Stong1968]. Anderson, Brown and Peterson determine the structure of modulo the ideal consisting of torsion mapping monomorphically into unoriented cobordism. According to [Laures2003], the multiplicative structure of this ideal is still not completely known.
Generators
, generated by a point.
, generated by , the circle with the "antiperiodic" spin structure.
, generated by .
.
, generated by , the Kummer surface.
.
, generated by quaternionic projective space, and a generator which equals 1/4 of the square of the Kummer surface.
By the above theorem of Anderson, Brown and Peterson there exist manifolds
- of dimension if is even,
- of dimension if is odd,
such that the characteristic numbers , and are odd. For odd, let be a spin nullbordism of , and let (using an orientation-reversing automorphism of ). If all are even, one can choose to be a product of quaternionic projective spaces.
Then a basis for is given by
- .
A basis for is given by
- with and even,
- with even,
- with odd,
- with and odd
- ,
- .
4 Further topics
Relation to framed bordism: The image of framed bordism is 0 except in dimensions where it is .
The kernel of the map from spin to oriented bordism is in dimensions and only and is the part generated by framed manifolds. It is the ideal generated by the non-trivial class of degree 1.
The image in unoriented bordism is all classes for which the characteristic numbers divisible by and are zero.
5 References
- [Anderson&Brown&Peterson1966] D. W. Anderson, E. H. Brown and F. P. Peterson, Spin cobordism, Bull. Amer. Math. Soc. 72 (1966), 256–260. MR0190939 (32 #8349) Zbl 0156.21605
- [Anderson&Brown&Peterson1967] D. W. Anderson, E. H. Brown and F. P. Peterson, The structure of the Spin cobordism ring, Ann. of Math. (2) 86 (1967), 271–298. MR0219077 (36 #2160) Zbl 0156.21605
- [Laures2003] G. Laures, An splitting of spin bordism, Amer. J. Math. 125 (2003), no.5, 977–1027. MR2004426 (2004g:55007) Zbl 1058.55001
- [Lawson&Michelsohn1989] H. B. Lawson and M. Michelsohn, Spin geometry, Princeton University Press, Princeton, NJ, 1989. MR1031992 (91g:53001) Zbl 0801.58017
- [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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