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].
2 Invariants
The spin bordism class of a manifold is detected by -cohomology (Stiefel-Whitney) and KO-theory (Pontryagin) characteristic numbers.
A spin structure on a closed -manifold induces a KO-orientation , so that we can evaluate polynomials in the KO-Pontryagin classes to get characteristic numbersThere is an interpretation of these characteristic numbers using index theory:
A feature of Spin manifolds is that they possess Dirac operators, . The (Clifford-linear) Dirac operator can be considered as a representative of the fundamental class , see [Atiyah1970] and [Higson&Roe2000]. The characteristic numbers above can then be defined as the indices of the (Clifford-linear) Dirac operators obtained by twisting with the corresponding vector bundles. The easiest case is the non-twisted one: is the trivial bundle, and taking the index of the Dirac operator defines an element of when is n-dimensional. This gives rises to a ring homomorphism often called the -invariant:
See [Lawson&Michelsohn1989]. For divisible by 4, this invariant equals (up to possibly a factor of 2) the -genus of .
3 Classification
3.1 MSpin away from the prime 2 and at the prime 2
After inverting 2 the map of Thom spectra becomes a homotopy equivalence. Thus there is no odd torsion in the spin cobordism groups, and all summands are in degrees divisible by 4.
Using the Thom isomorphism we get for each multi-index 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 3.1 [Anderson&Brown&Peterson1967]. There are classes such that there is a 2-local homotopy equivalence
3.2 Consequences
From this one can compute the additive structure completely. 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.
3.3 Ring structure
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.
4 Generators
4.1 Low dimensions
, 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.
4.2 Generators in all dimensions as given by the classification
For a multi-index where all , we set and . By the theorem of Anderson, Brown and Peterson below 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 ). By [Stong1966] there exist manifolds such that . 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
- ,
- .
5 Further topics
5.1 Relationship with other bordism groups
- Framed bordism the image of is 0 unless or when it is and detected by the -invariant. Explicit generators are with and .
- Oriented bordism: the kernel of lies in dimensions and .It is a vector space with a basis , even, and , odd where and . It is also the ideal generated by the non-trivial class of . The cokernel is a finite -torsion group which is trivial if and only of or equivalently , , , , and . [Milnor1965] computed .
- Unoriented bordism: the image of is all bordism classes for which the characteristic numbers divisible by and are zero. A basis for the image consists of the , even, the , odd and the . The first occurs in dimension . The image is trivial for . In even dimensions it is additionally trivial for and . In odd dimensions it is trivial for and also for and . Otherwise the image is non trivial. Indeed, there are in all dimensions .
6 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
- [Atiyah1970] M. F. Atiyah, Global theory of elliptic operators, (1970), 21–30. MR0266247 (42 #1154) Zbl 0193.43601
- [Higson&Roe2000] N. Higson and J. Roe, Analytic -homology, Oxford University Press, Oxford, 2000. MR1817560 (2002c:58036) Zbl 1146.19004
- [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
- [Milnor1965] J. W. Milnor, Remarks concerning spin manifolds, in Differential and Combinatorial Topology, a Symposium in Honor of Marston Morse, (1965) 55–62. MR0180978 (31 #5208) Zbl 0132.19602
- [Stong1966] R. E. Stong, Relations among characteristic numbers. II, Topology 5 (1966), 133–148. MR0192516 (33 #741) Zbl 0142.40902
- [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010