String bordism
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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 Graeme Segal, elliptic cohomoloy, Seminaire Bourbaki 40e, 1987-88, 695). | + | 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 Graeme Segal, elliptic cohomoloy, Seminaire Bourbaki 40e, 1987-88, 695 \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 Algebraic topology and modular forms. Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 291--317). 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. | + | from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms. This spectrum was developed by Mike Hopkins (see Algebraic topology and modular forms. Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 291--317)\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. |
</wikitex> | </wikitex> | ||
Revision as of 14:54, 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 V. Giambalvo, On -cobordism, [Giambalvo1971] Illinois J. Math. 15 1971 533–541 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 Graeme Segal, elliptic cohomoloy, Seminaire Bourbaki 40e, 1987-88, 695 [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 Algebraic topology and modular forms. Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 291--317)[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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