String bordism

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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 below.

\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.

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-Pries2009]). A more explicit model for this extension can be found in [Meinrenken2003].

3 The bordism groups

The additive structure of the bordism groups is not fully determined yet. It is known that only 2 and 3 torsion appears (see [Giambalvo1971]) and the 3 torsion is annihalated by multiplication with 3 (see [Hovey1997]). Moreover, the bordism groups \Omega_{k}^{String} are finite for k=1,2,3 mod 4.

Clearly, since BO\!\left< 8 \right> is 7-connected the first 6 bordism groups coincide with the framed bordism groups. 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 [Stong1963]:

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

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 [Giambalvo1969].

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-algebra structure is free (see [Laures2003a]).

4.3 K(n)-local computations

For K=K(n) at p=2 one has an exact sequence of Hopf algebras (see [Kitchloo&Laures&Wilson2004a])

\displaystyle  \xymatrix{K_* \ar[r] & K_*K({\mathbb Z}/2,2)\ar[r]& K_*K({\mathbb Z},3)\ar[r]& K_*BString \ar[r]&K_*BSpin \ar[r]& K_*  K({\mathbb Z},4)\ar[r]^2& K({\mathbb Z},4)\ar[r]&K_*}

which is induced by the obvious geometric maps. For n=2 it algebraically reduces to the split exact sequence of Hopf algebras (see [Kitchloo&Laures&Wilson2004b])

\displaystyle \xymatrix{K(2)_* \ar[r] & K(2)_*K({\mathbb Z},3)\ar[r]& K(2)_*BString \ar[r]&K(2)_*BSpin \ar[r]& K(2)_*}.

4.4 Computations with respect to general complex oriented theories

Ando, Hopkins and Strickland investigated the homology ring E_*BString for even periodic multiplicative cohomology theories E. Even periodic theories are complex orientable which means that E^0({\mathbb C}P^\infty) carries a formal group. The description of E_*BString is in terms of formal group data.

In [Ando&Hopkins&Strickland2001a] first the analogous complex problem is studied. The group O\left< 8 \right>=String has a complex relative U\left< 6 \right> which is defined in the same way by killing the third homotopy group of SU. Consider the map

\displaystyle  (1-L_1)(1-L_2)(1-L_3):  \xymatrix{ {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty\ar[r]& BU}

where the L_i are the canonical line bundles over the individual factors. Since the first two Chern classes of the virtual bundle vanish the map lifts to BU\left<6 \right>. If we choose a complex orienatation the lift gives a class f in the cohomology ring

\displaystyle (E\wedge BU\left<6 \right>_+)^0( {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty )\cong  E_0BU\left<6 \right> [\![x_1,x_2,x_3]\!]

with c_i(L_i)=x_i. The power series f satisfies the following identities:

\displaystyle  \begin{aligned} f(0,0,0)&=1\\ f(x_1,x_2,x_3)&=f(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)});\; \sigma \in \Sigma_3\\ f(x_1,x_2,x_3)f(x_0,x_1+_Fx_2,x_3)&=f(x_0+_Fx_1,x_2,x_3)f(x_0,x_1,x_2). \end{aligned}

Hence, the power series is an example a symmetric 3-cocycle or a cubical structure on the formal group law. In fact the main result of [Ando&Hopkins&Strickland2001a] is that it is the universal example of such a structure. Explicitly, this means that the commutative ring E_0 BU\left <6 \right> is freely generated by the coefficients of f subject to the relations given by the 3 equations above.

The real version of this result has not been published yet by the three authors. Using the diagram

\displaystyle  \xymatrix{K{\mathbb Z},3)\ar[r]\ar[d]^=&BU\left< 6 \right>\ar[r]\ar[d]& BSU\ar[d]\\ K({\mathbb Z},3)\ar[r]& BString \ar[r]&BSpin}

and the results for K(2)_*BString described above they conjecture that E_0BString is the same quotient subject to the additional relation

\displaystyle  f(x_1,x_2,x_3)=1 \mbox{ for } x_1+_Fx_2+_Fx_3=0.

5 The structure of the spectrum

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]). For p=3 there is a spectrum Y with 3 cells in even dimensions such that MString\wedge Y splits into a sum of suspensions of BP. For p=2 it is hoped that the spectrum tmf splits off which is explained below.

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 ([Hopkins2002]). This map is also called the \sigma-orientation and is 15-connected (see [Hill2008]). The spectrum TMF was developed by Goerss, Hopkins and Miller. 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 References

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