String bordism

Revision as of 17:10, 31 March 2011

1 Introduction

${{Stub}} == Introduction == <wikitex>; $String$-bordism or $O\!\left< 8 \right>$-bordism is a special case of a [[B-Bordism|B-bordism]]. It comes from the tower of fibrations below. $$ \xymatrix{BO\!\left< 8 \right>\ar[d] & \ BSpin \ar[d]\ar[r]^{p_1/2}&K({\mathbb Z},4) \ BSO\ar[d]\ar[r]^{w_2}& K({\mathbb Z}/2,2) \ BO\ar[r]^{w_1}& K({\mathbb Z}/2,1) } $$ In each step the lowest homotopy group is killed by the map into the Eilenberg-MacLane spaces. In particular, $BO\left< 8 \right>$ is the homotopy fibre of the map from $BSpin$ given by half of the first Pontryagin class. The name $String$-group is due to Haynes Miller and will be explained below. </wikitex> == The String group == <wikitex>; 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}. </wikitex> == Low dimensional generators == <wikitex>; The following computations of $\Omega_*^{String}$ for $* \leq 16$ comes from \cite{Giambalvo1971|p. 538}. * For $i \leq 6$ the natural map $\Omega_i^{fr} \to \Omega_i^{String}$ is an isomorphism. See [[Framed bordism|framed bordism]] for these groups. * $\Omega_7^{String} = 0$. * $\Omega_8^{String} \cong \Zz \oplus \Zz/2$, generated by the [[Exotic spheres|exotic 8-sphere]] for the 2-torsion and a certain [[Bott manifold]]: see \cite{Laures2004}. * $\Omega_9^{String} \cong \Theta_9/bP_{10} \cong (\Zz/2)^2$, generated by [[Exotic spheres|exotic 9-spheres]]. * $\Omega_{10}^{String} \cong \Zz/6$, generated by an [[Exotic spheres|exotic 10-sphere]]. * $\Omega_{11}^{String} = 0$. * $\Omega_{12}^{String} \cong \Zz$, generated by a 5-connected manifold with signature \times 992$. * $\Omega_{13}^{String} = 0$. * $\Omega_{14}^{String} \cong \Zz/2$, generated by the [[Exotic spheres|exotic 14-sphere]]. * $\Omega_{15}^{String} \cong \Zz/2$, genreated by the [[Exotic spheres|exotic 15-sphere]]. * $\Omega_{16}^{String} \cong \Zz^2$. </wikitex> == Homology calculations == ===Singular homology=== <wikitex>; The cohomology ring $H^*(BString,{\mathbb Z}/p)$ has been computed for $p=2$ by Stong in \cite{Stong63}: $$ H^*(BString)\cong H^*(K({\mathbb Z},4))/Sq^2(\iota) \otimes {\mathbb Z}/2[\theta_i|\sigma_2(i+1)>4].$$ Here, $\sigma_2$ is the number of digits in the duadic decomposition and the $\theta_i$ come from the cohomology of $BO$ and coincide with the Stiefel-Whitney up to decomposables. For odd $p$ the corresponding result has been obtained by Giambalvo \cite{Giambalvo69}. </wikitex> ===K(1)-local computations=== <wikitex>; $K(1)$ locally $MString$ coincides with $MSpin$ and decomposes into a wedge of copies of $KO$. However, it is not an algebra over $KO$. Its multiplicative structure can be read off the formula $$ L_{K(1)}MString \cong T_\zeta \wedge \bigwedge TS^0.$$ Here, $\zeta\in \pi_{-1}L_{K(1)}S^0$ is a generator, $T_\zeta$ is the $E_\infty $ cone over $\zeta$ and $TS^0$ is the free $E_\infty$ spectrum generated by the sphere. ===$K(2)$-local computations=== ===Computations with respect to complex oriented theories=== <wikitex>; 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}). Indeed, localized at a prime $p>3$, string bordism splits additively into a sum of suspensions of $BP$, although the ring structure is different (see \cite{Hovey2008}). The first 16 bordism groups has been computed by Giambalvo, at the prime 3 the first 32 bordism groups can be found in \cite{Hovey&Ravenel1995}. Further calculations have been done in \cite{Mahowald&Gorbounov1995}. </wikitex> == The structure of the spectrum== <wikitex>; </wikitex> == The Witten genus == <wikitex>; 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 k$-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{Segal1988}.) The Witten genus can be refined to a map of structured ring spectra $$W: MString \longrightarrow TMF$$ from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms (\cite{Ando&Hopkins&Rezk2006}). This map is also called the $\sigma$-orientation and is 15-connected (see \cite{Hill2008}). The spectrum $TMF$ was developed by Mike Hopkins and Haynes Miller (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 (see \cite{Hopkins&Mahowald2002}.) </wikitex> == Characteristic numbers and index theory== <wikitex>; == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] [[Category:Bordism]]String$-bordism or $O\!\left< 8 \right>$-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, $BO\left< 8 \right>$ is the homotopy fibre of the map from $BSpin$ given by half of the first Pontryagin class. The name $String$-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 $String(n)$ 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 [Schommer-Pries2010]). An 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. Clearly, since $BO\!\left< 8 \right>$ 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]) and the 3 torsion is annihalated by multiplication with 3 (see [Hovey97]). The bordism groups $\Omega_{k}^{String}$ are finite for $k=1,2,3$ mod 4.

The first 16 bordism groups have been computed by Giambalvo [Giambalvo1971, p. 538]:

• $\Omega_7^{String} = 0$.
• $\Omega_8^{String} \cong \Zz \oplus \Zz/2$, generated by the exotic 8-sphere for the 2-torsion and a certain Bott manifold: see [Laures2004].
• $\Omega_9^{String} \cong \Theta_9/bP_{10} \cong (\Zz/2)^2$, generated by exotic 9-spheres.
• $\Omega_{10}^{String} \cong \Zz/6$, generated by an exotic 10-sphere.
• $\Omega_{11}^{String} = 0$.
• $\Omega_{12}^{String} \cong \Zz$, generated by a 5-connected manifold with signature $8 \times 992$.
• $\Omega_{13}^{String} = 0$.
• $\Omega_{14}^{String} \cong \Zz/2$, generated by the exotic 14-sphere.
• $\Omega_{15}^{String} \cong \Zz/2$, genreated by the exotic 15-sphere.
• $\Omega_{16}^{String} \cong \Zz^2$.

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 $H^*(BString,{\mathbb Z}/p)$ has been computed for $p=2$ by Stong in [Stong63]:

$$\displaystyle H^*(BString)\cong H^*(K({\mathbb Z},4))/Sq^2(\iota) \otimes {\mathbb Z}/2[\theta_i|\sigma_2(i+1)>4].$$

Here, $\sigma_2$ is the number of digits in the duadic decomposition and the $\theta_i$ come from the cohomology of $BO$ and coincide with the Stiefel-Whitney up to decomposables. For odd $p$ the corresponding result has been obtained by Giambalvo [Giambalvo69].

4.2 K(1)-local computations

$K(1)$ locally $MString$ coincides with $MSpin$ and decomposes into a wedge of copies of $KO$. However, it is not an algebra over $KO$. Its multiplicative structure for $p=2$ can be read off the formula

$$\displaystyle L_{K(1)}MString \cong T_\zeta \wedge \bigwedge TS^0.$$

Here, $\zeta\in \pi_{-1}L_{K(1)}S^0$ is a generator, $T_\zeta$ is the $E_\infty$ cone over $\zeta$ and $TS^0$ is the free $E_\infty$ spectrum generated by the sphere. In particular, its $\theta$-algabra structure is free (see [Lau03]).

1 $K(2)$-local computations

2 Computations with respect to general complex oriented theories

5 The structure of the spectrum

Indeed, localized at a prime $p>3$, string bordism splits additively into a sum of suspensions of $BP$, although the ring structure is different (see [Hovey2008]). 

6 The Witten genus

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 [Segal1988].) The Witten genus can be refined to a map of structured ring spectra

$$\displaystyle W: MString \longrightarrow TMF$$

from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms ([Ando&Hopkins&Rezk2006]). This map is also called the $\sigma$-orientation and is 15-connected (see [Hill2008]). The spectrum $TMF$ was developed by Mike Hopkins and Haynes Miller (see [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 (see [Hopkins&Mahowald2002].)

7 Characteristic numbers and index theory

References

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• [Hill2008] M. A. Hill, The String bordism of $BE_8$ and $BE_8\times BE_8$ 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 $BO[8]$, Recent progress in homotopy theory (Baltimore, MD, 2000), Contemp. Math., 293 (2002), 89–110. MR1887530 (2003b:55007) Zbl 1012.57041
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• [Hovey97] Template:Hovey97
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• [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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