String bordism
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== Additive structure == | == Additive structure == | ||
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− | The additive structure of the bordism groups is not fully determined yet. Clearly, since $BO\left< 8 \right>$ is 7-connected the first 6 bordism groups coincide with the [[Framed bordism|framed bordism groups]]. It is known that only 2 and 3 torsion appears (see | + | The additive structure of the bordism groups is not fully determined yet. Clearly, since $BO\left< 8 \right>$ is 7-connected the first 6 bordism groups coincide with the [[Framed bordism|framed bordism groups]]. It is known that only 2 and 3 torsion appears (see \cite{Giambalvo1971} also for the first 16 bordism groups). |
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== The Witten genus == | == The Witten genus == | ||
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− | At the end of the 80s Ed Witten were studying the $S^1$-equivariant index of the Dirac operator on a loop space of a $4k$-dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space $LM$ this would mean that $M$ is $String$. Witten carried the Atiyah-Segal formula for the index over to this infinite dimensional setting and obtained an integral modular form of weight $k$. Nowadays this is called the Witten genus (see | + | At the end of the 80s Ed Witten were studying the $S^1$-equivariant index of the Dirac operator on a loop space of a $4k$-dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space $LM$ this would mean that $M$ is $String$. Witten carried the Atiyah-Segal formula for the index over to this infinite dimensional setting and obtained an integral modular form of weight $k$. Nowadays this is called the Witten genus (see \cite{Segal1987}). |
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==Characteristic numbers== | ==Characteristic numbers== | ||
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The Witten genus can be refined to a map of structured ring spectra | The Witten genus can be refined to a map of structured ring spectra | ||
$$W: MString \longrightarrow TMF$$ | $$W: MString \longrightarrow TMF$$ | ||
− | from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms. This spectrum was developed by Mike Hopkins (see | + | from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms. This spectrum was developed by Mike Hopkins (see \cite{Hopkins2002}. It is supposed to play the same role for $String$-bordism as $KO$-theory does for $Spin$-bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map $W$ gives characteristic numbers which together with $KO$ and Stiefel-Whitney numbers are conjectured to determine the $String$ bordism class. Moreover, $tmf$ is supposed to be a direct summand of $MString$ as the orientation map $W$ is shown to be surjective in homotopy. |
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Revision as of 14:57, 12 February 2010
Contents |
1 Introduction
-bordism or -bordism is a special case of a B-bordism. It comes from the tower of fibrations
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. There are various models for the String group.
2 Additive structure
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] also for the first 16 bordism groups). ;
3 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 [Segal1987]).;
4 Characteristic numbers
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. This spectrum was developed by Mike Hopkins (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.
5 References
- [Giambalvo1971] V. Giambalvo, On -cobordism, Illinois J. Math. 15 (1971), 533–541. MR0287553 (44 #4757) Zbl 0221.57019
- [Hopkins2002] M. J. Hopkins, Algebraic topology and modular forms, (2002), 291–317. MR1989190 (2004g:11032) Zbl 1031.55007
- [Segal1987] Template:Segal1987
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