String bordism
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[edit] 1 Introduction
-bordism or -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, 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.
[edit] 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-Pries2009]). A more explicit model for this extension can be found in [Meinrenken2003].
[edit] 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 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 an exotic 13-sphere.
- , generated by the exotic 14-sphere.
- , genreated by an 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].
[edit] 4 Homology calculations
[edit] 4.1 Singular homology
The cohomology ring has been computed for by Stong in [Stong1963]:
Here, is the number of ones in the duadic decomposition and the come from the cohomology of and coincide with the Stiefel-Whitney up to decomposables.
From Stong's proof it follows that is surjective, and is a polynomial algebra.
For odd the corresponding result has been obtained by Giambalvo [Giambalvo1969].
[edit] 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
Here, is a generator, is the cone over and is the free spectrum generated by the sphere. In particular, its -algebra structure is free (see [Laures2003a]).
[edit] 4.3 K(n)-homology computations
For Morava at one has an exact sequence of Hopf algebras (see [Kitchloo&Laures&Wilson2004a])
which is induced by the obvious geometric maps. For it algebraically reduces to the split exact sequence of Hopf algebras (see [Kitchloo&Laures&Wilson2004b])
[edit] 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.
In [Ando&Hopkins&Strickland2001a] first the analogous complex problem is studied. The group has a complex relative which is defined in the same way by killing the third homotopy group of . Consider the map
where the are the canonical line bundles over the individual factors. Since the first two Chern classes of the virtual bundle vanish the map lifts to . If we choose a complex orienatation the lift gives a class in the cohomology ring
with . The power series satisfies the following identities:
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 is freely generated by the coefficients of 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
and the results for described above they conjecture that is the same quotient subject to the additional relation
[edit] 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.
Localized at a prime , the string bordism ring injects as a nonpolynomial subring of the oriented bordism ring. (A toy model worth bearing in mind is the inclusionTex syntax error.) Thus, Pontrjagin numbers suffice to distinguish elements of . In fact, a set generates as a -algebra if:
- For each integer , there is an element of such that:
- For each pair of integers , there is an element of such that:
where is the Milnor number, the characteristic number corresponding to the power sum polynomial of the Pontrjagin roots , and is the characteristic number corresponding to the symmetric polynomial (see [McTague2014, Theorem 4]).
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[edit] 6 The Witten genus
At the end of the 80s Ed Witten was 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
from the Thom spectrum of String bordism to the spectrum of topological modular forms ([Hopkins2002]). This map is also called the -orientation and is 15-connected (see [Hill2008]). The spectrum was developed by Goerss, Hopkins and Miller. 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].)
[edit] 7 References
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- [Giambalvo1969] V. Giambalvo, The cohomology of , Proc. Amer. Math. Soc. 20 (1969), 593–597. MR0236913 (38 #5206) Zbl 0176.52601
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