String bordism
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1 Introduction
-bordism or -bordism is a special case of a B-bordism. It comes from the tower of fibrations below.
In each step the lowest homotopy group is killed by the map into the Eilenberg-MacLane spaces. In particular, is the homotopy fibre of the map from given by half of the first Pontryagin class. The name -group is due to Haynes Miller and will be explained below.
2 The String group
There are various models for the String group. However, since its first three homotopy groups vanish it can not be realized as a finite dimensional Lie group. There are homotopy theoretic constructions which involve path spaces. A more geometric way to think about which involves only finite dimensional manifolds is the following: the String group fibers over the Spin group with fibre . One may think of as the realization of viewed as a smooth category with only one object. This way, the space appears as the realization of a smooth 2-group extension of by the finite dimensional Lie groupoid (see [Schommer-Pries2010]). A more explicit model for this extension can be found in [Meinrecken03].
3 The bordism groups
The additive structure of the bordism groups is not fully determined yet. It is known that only 2 and 3 torsion appears (see [Giambalvo1971]) and the 3 torsion is annihalated by multiplication with 3 (see [Hovey97]). Moreover, the bordism groups are finite for mod 4.
Clearly, since is 7-connected the first 6 bordism groups coincide with the framed bordism groups. The first 16 bordism groups have been computed by Giambalvo [Giambalvo1971, p. 538]:
- .
- , generated by the exotic 8-sphere for the 2-torsion and a certain Bott manifold: see [Laures2004].
- , generated by exotic 9-spheres.
- , generated by an exotic 10-sphere.
- .
- , generated by a 5-connected manifold with signature .
- .
- , generated by the exotic 14-sphere.
- , genreated by the exotic 15-sphere.
- .
At the prime 3 the first 32 bordism groups can be found in [Hovey&Ravenel1995]. Further calculations have been done in [Mahowald&Gorbounov1995].
4 Homology calculations
4.1 Singular homology
The cohomology ring has been computed for by Stong in [Stong63]:
Here, is the number of digits in the duadic decomposition and the come from the cohomology of and coincide with the Stiefel-Whitney up to decomposables. For odd the corresponding result has been obtained by Giambalvo [Giambalvo69].
4.2 K(1)-local computations
locally coincides with and decomposes into a wedge of copies of . However, it is not an algebra over . Its multiplicative structure for can be read off the formula
Here, is a generator, is the cone over and is the free spectrum generated by the sphere. In particular, its -algabra structure is free (see [Lau03]).
4.3 K(n)-local computations
For at one has an exact sequence of Hopf algebras (see [KLW04a])
which is induced by the obvious geometric maps. For it algebraically reduces to the split exact sequence of Hopf algebras (see [KLW04b])
4.4 Computations with respect to general complex oriented theories
Ando, Hopkins and Strickland...
5 The structure of the spectrum
Localized at a prime , string bordism splits additively into a sum of suspensions of , although the ring structure is different (see [Hovey2008]). For there is a spectrum with 3 cells in even dimensions such that splits into a sum of suspensions of . For it is hoped that the spectrum splits off which is explained below.
6 The Witten genus
At the end of the 80s Ed Witten were studying the -equivariant index of the Dirac operator on a loop space of a -dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space this would mean that is . Witten carried the Atiyah-Segal formula for the index over to this infinite dimensional setting and obtained an integral modular form of weight . Nowadays this is called the Witten genus (see [Segal1988].) The Witten genus can be refined to a map of structured ring spectra
from the Thom spectrum of String bordism to the spectrum of topological modular forms ([Ando&Hopkins&Rezk2006]). This map is also called the -orientation and is 15-connected (see [Hill2008]). The spectrum was developed by Mike Hopkins and Haynes Miller (see [Hopkins2002].) It is supposed to play the same role for -bordism as -theory does for -bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map gives characteristic numbers which together with and Stiefel-Whitney numbers are conjectured to determine the bordism class. Moreover, is supposed to be a direct summand of as the orientation map is shown to be surjective in homotopy (see [Hopkins&Mahowald2002].)
7 References
- [Ando&Hopkins&Rezk2006] Template:Ando&Hopkins&Rezk2006
- [Giambalvo1971] V. Giambalvo, On -cobordism, Illinois J. Math. 15 (1971), 533–541. MR0287553 (44 #4757) Zbl 0221.57019
- [Giambalvo69] Template:Giambalvo69
- [Hill2008] M. A. Hill, The String bordism of and through dimension 14, (2008). Available at the arXiv:arXiv:0807.2095v1.
- [Hopkins&Mahowald2002] M. Mahowald and M. Hopkins, The structure of 24 dimensional manifolds having normal bundles which lift to , Recent progress in homotopy theory (Baltimore, MD, 2000), Contemp. Math., 293 (2002), 89–110. MR1887530 (2003b:55007) Zbl 1012.57041
- [Hopkins2002] M. J. Hopkins, Algebraic topology and modular forms, (2002), 291–317. MR1989190 (2004g:11032) Zbl 1031.55007
- [Hovey&Ravenel1995] M. A. Hovey and D. C. Ravenel, The -connected cobordism ring at , Trans. Amer. Math. Soc. 347 (1995), no.9, 3473–3502. MR1297530 (95m:55008) Zbl 0852.55008
- [Hovey2008] M. Hovey, The homotopy of and at large primes, Algebr. Geom. Topol. 8 (2008), no.4, 2401–2414. MR2465746 (2009h:55002) Zbl 1165.55001
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- [Laures2004] G. Laures, -local topological modular forms, Invent. Math. 157 (2004), no.2, 371–403. MR2076927 (2005h:55003) Zbl 1078.55010
- [Mahowald&Gorbounov1995] M. Mahowald and V. Gorbounov, Some homotopy of the cobordism spectrum , Homotopy theory and its applications (Cocoyoc, 1993), Amer. Math. Soc. (1995), 105–119. MR1349133 (96i:55010) Zbl 0840.55002
- [Meinrecken03] Template:Meinrecken03
- [Schommer-Pries2010] Template:Schommer-Pries2010
- [Segal1988] G. Segal, Elliptic cohomology (after Landweber-Stong, Ochanine, Witten, and others), Séminaire Bourbaki, Vol. 1987/88, Astérisque No. 161-162 (1988), Exp. No. 695, 4, (1989) 187–201. MR992209 (91b:55005) Zbl 0686.55003
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