String bordism
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
-bordism or
-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) }](/images/math/3/7/b/37bd1f92c9a30f2b9da4996d3a8aa181.png)
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.
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 which involves only finite dimensional manifolds is the following: the String group fibers over the Spin group with fibre
. One may think of
as the realization of
viewed as a smooth category with only one object. This way, the
space
appears as the realization of a smooth 2-group extension of
by the finite dimensional Lie groupoid
(see [Schommer-Pries2010]). A 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. It is known that only 2 and 3 torsion appears (see [Giambalvo1971]) and the 3 torsion is annihalated by multiplication with 3 (see [Hovey97]). Moreover, the bordism groups are finite for
mod 4.
Clearly, since 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]:
-
.
-
, generated by the exotic 8-sphere for the 2-torsion and a certain Bott manifold: see [Laures2004].
-
, generated by exotic 9-spheres.
-
, generated by an exotic 10-sphere.
-
.
-
, generated by a 5-connected manifold with signature
.
-
.
-
, generated by the exotic 14-sphere.
-
, genreated by the exotic 15-sphere.
-
.
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 has been computed for
by Stong in [Stong63]:
![\displaystyle H^*(BString)\cong {\mathbb Z}/2[\theta_i|\sigma_2(i+1)>4]\otimes H^*(K({\mathbb Z},4))/Sq^2(\iota).](/images/math/1/6/6/1664d6ce54e6548005e8bdcb8febb4bb.png)
Here, is the number of digits in the duadic decomposition and the
come from the cohomology of
and coincide with the Stiefel-Whitney up to decomposables. For odd
the corresponding result has been obtained by Giambalvo [Giambalvo69].
4.2 K(1)-local computations
locally
coincides with
and decomposes into a wedge of copies of
. However, it is not an algebra over
. Its multiplicative structure for
can be read off the formula
![\displaystyle L_{K(1)}MString \cong T_\zeta \wedge \bigwedge TS^0.](/images/math/7/c/0/7c041893bb4af4d93ccbeb9504ad6e25.png)
Here, is a generator,
is the
cone over
and
is the free
spectrum generated by the sphere. In particular, its
-algabra structure is free (see [Lau03]).
4.3 K(n)-local computations
For at
one has an exact sequence of Hopf algebras (see [KLW04a])
![\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_*}](/images/math/a/e/c/aec9eeebe1b3ea17211973d679e21b2b.png)
which is induced by the obvious geometric maps. For it algebraically reduces to the split exact sequence of Hopf algebras (see [KLW04b])
![\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)_*}.](/images/math/8/d/3/8d3e325bb23de7bca738ff651bce7d49.png)
4.4 Computations with respect to general complex oriented theories
Ando, Hopkins and Strickland investigated the homology ring for even periodic multiplicative cohomology theories
. Even periodic theories are complex orientable which means that
carries a formal group. The description of
is in terms of formal group data.
![O\left< 8 \right>=String](/images/math/a/4/5/a452d6bda94595d67ebb7a9076ec9d39.png)
Tex syntax errorwhich is defined in the same manner by killing the third homotopy group of
![SU](/images/math/4/f/c/4fca10dc3542256f3be70b5a195ca39a.png)
![\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}](/images/math/d/0/7/d0706e7dffb5e329157e970a95141e54.png)
![L_i](/images/math/d/8/a/d8a751a575054243dba03c89bee8107c.png)
Tex syntax error. If we choose a complex orienatation the lift gives a class
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
ring
Tex syntax error
with . The power series
satisfies the following identities:
\begin{eqnarray*}
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{eqnarray*}
Tex syntax erroris freely generated by the coefficients of
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
The real version of this result has not been published yet by the three authors. Using the diagram
Tex syntax error
and the results for described above that
is the same quotient subject to the additional relation
![\displaystyle f(x_1,x_2,x_3)=1](/images/math/e/5/4/e54d95a94979e5d96ccff360002b4c70.png)
Tex syntax error
5 The structure of the spectrum
Localized at a prime , string bordism splits additively into a sum of suspensions of
, although the ring structure is different (see [Hovey2008]). For
there is a spectrum
with 3 cells in even dimensions such that
splits into a sum of suspensions of
. For
it is hoped that the spectrum
splits off which is explained below.
6 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 [Segal1988].)
The Witten genus can be refined to a map of structured ring spectra
![\displaystyle W: MString \longrightarrow TMF](/images/math/d/2/d/d2d109f54a8e983e2bb70d50bfde7391.png)
from the Thom spectrum of String bordism to the spectrum of topological modular forms ([Ando&Hopkins&Rezk2006]). This map is also called the
-orientation and is 15-connected (see [Hill2008]). The spectrum
was developed by Mike Hopkins and Haynes Miller (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 (see [Hopkins&Mahowald2002].)
7 References
- [AHS02] Template:AHS02
- [AHS03] Template:AHS03
- [Ando&Hopkins&Rezk2006] Template:Ando&Hopkins&Rezk2006
- [Giambalvo1971] V. Giambalvo, On
-cobordism, Illinois J. Math. 15 (1971), 533–541. MR0287553 (44 #4757) Zbl 0221.57019
- [Giambalvo69] Template:Giambalvo69
- [Hill2008] M. A. Hill, The String bordism of
and
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
, Recent progress in homotopy theory (Baltimore, MD, 2000), Contemp. Math., 293 (2002), 89–110. MR1887530 (2003b:55007) Zbl 1012.57041
- [Hopkins2002] M. J. Hopkins, Algebraic topology and modular forms, (2002), 291–317. MR1989190 (2004g:11032) Zbl 1031.55007
- [Hovey&Ravenel1995] M. A. Hovey and D. C. Ravenel, The
-connected cobordism ring at
, Trans. Amer. Math. Soc. 347 (1995), no.9, 3473–3502. MR1297530 (95m:55008) Zbl 0852.55008
- [Hovey2008] M. Hovey, The homotopy of
and
at large primes, Algebr. Geom. Topol. 8 (2008), no.4, 2401–2414. MR2465746 (2009h:55002) Zbl 1165.55001
- [Hovey97] Template:Hovey97
- [KLW04a] Template:KLW04a
- [KLW04b] Template:KLW04b
- [Lau03] Template:Lau03
- [Laures2004] G. Laures,
-local topological modular forms, Invent. Math. 157 (2004), no.2, 371–403. MR2076927 (2005h:55003) Zbl 1078.55010
- [Mahowald&Gorbounov1995] M. Mahowald and V. Gorbounov, Some homotopy of the cobordism spectrum
, Homotopy theory and its applications (Cocoyoc, 1993), Amer. Math. Soc. (1995), 105–119. MR1349133 (96i:55010) Zbl 0840.55002
- [Meinrecken03] Template:Meinrecken03
- [Schommer-Pries2010] Template:Schommer-Pries2010
- [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
- [Stong63] Template:Stong63