String bordism

From Manifold Atlas
Revision as of 18:01, 12 February 2010 by Diarmuid Crowley (Talk | contribs)
Jump to: navigation, search

Contents

1 Introduction

String-bordism or O\!\left< 8 \right>-bordism is a special case of a B-bordism. It comes from the tower of fibrations

\displaystyle  \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. 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 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] also for the first 16 bordism groups.)

3 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].)

4 Characteristic numbers

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. This spectrum was developed by Mike Hopkins (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].)

5 References

This page has not been refereed. The information given here might be incomplete or provisional.

Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox