String bordism
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== The String group == | == The String group == | ||
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− | 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 $String(n)$ which involves only finite dimensional manifolds is the following: the String group fibers over the Spin group with fibre $K({\mathbb Z},2)$. One may think of $ K({\mathbb Z},2)$ as the realization of $S^1$ viewed as a smooth category with only one object. This way, the $A_\infty$ space $String(n)$ appears as the realization of a smooth 2-group extension of $Spin(n)$ by the finite dimensional Lie groupoid $S^1$ (see \cite{Schommer-Pries2010}). An explicit model for this extension can be found in \cite{Meinrecken03}. | + | 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 $String(n)$ which involves only finite dimensional manifolds is the following: the String group fibers over the Spin group with fibre $K({\mathbb Z},2)$. One may think of $ K({\mathbb Z},2)$ as the realization of $S^1$ viewed as a smooth category with only one object. This way, the $A_\infty$ space $String(n)$ appears as the realization of a smooth 2-group extension of $Spin(n)$ by the finite dimensional Lie groupoid $S^1$ (see \cite{Schommer-Pries2010}). An more explicit model for this extension can be found in \cite{Meinrecken03}. |
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Revision as of 15:43, 31 March 2011
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Contents |
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]). An more explicit model for this extension can be found in [Meinrecken03].
3 Low dimensional generators
The following computations of for comes from [Giambalvo1971, p. 538].
- For the natural map is an isomorphism. See framed bordism for these groups.
- .
- , 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.
- .
4 Homology calculations
4.1 Singular homology
4.2 $K(1)$-local computations
4.3 $K(2)$-local computations
4.4 Computations with respect to complex oriented theories
The additive structure of the bordism groups is not fully determined yet. Clearly, since is 7-connected the first 6 bordism groups coincide with the framed bordism groups. It is known that only 2 and 3 torsion appears (see [Giambalvo1971]). Indeed, localized at a prime , string bordism splits additively into a sum of suspensions of , although the ring structure is different (see [Hovey2008]). The first 16 bordism groups has been computed by Giambalvo, at the prime 3 the first 32 bordism groups can be found in [Hovey&Ravenel1995]. Further calculations have been done in [Mahowald&Gorbounov1995].
5 The structure of the spectrum
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 Characteristic numbers and index theory
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
- [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
- [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