Spin bordism
Contents |
1 Introduction
The spin bordism groups of manifolds with spin structures are the homotopy groups of the Thom spectrum .
[Stong1968] [Laures2003] Preliminary results were by Novikov. The main calculation is due to Anderson, Brown, and Peterson [Anderson&Brown&Peterson1966] who showed that all torsion is of order 2, being of two types: that arising by products with a framed and that which maps monomorphically into unoriented cobordism. 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.
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.
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&Peterson1966] 3.1. There are classes such that there is a 2-local homotopy equivalence
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 ).
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 cobordism is 0 except in dimensions $8k+1,8k+2$ where it is $\Zz_2$.
The kernel of the map from spin to oriented bordism is in dimensions $8k + 1$ and $8k + 2$ 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 cobordism is all classes for which the characteristic numbers divisible by $w_1$ and $w_2$ are zero.
4.1 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
- [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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