# Does the existence of a string structure depend on a spin structure ? (and a generalization)

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## [edit] 1 Question

Let be a (stable) vector bundle. This has a classifying map .

A -structure on is (the vertical homotopy class of) a lift of the classifying map to a map . (For this is an orientation, a spin structure, a string structure respectively.)

Since for , the map factors through , a -structure induces a -structure, or, vice versa, this specific -structure can be lifted to a -structure.

**Question 1.1.**
Given a vector bundle and two -structures on it,
is it possible that one of the -structures can be lifted to a -structure and the other can't?

## [edit] 2 Answers

This is not possible for and , i.e. the question whether an oriented vector bundle admits a spin structure does not depend on the orientation, and the question whether a spin vector bundle admits a string structure does not depend on the spin structure. The reason in the first case is the obstruction to admitting a spin structure is the second Stiefel-Whitney class which is a homotopy invariant. For the second, more subtle point, of why the spin characteristic class does not depend upon the choice of spin structure see [Čadek&Crabb&Vanvzura2008, Defintion p.170].

The answer is however yes for all larger with . For example the existence of a -structure on a string vector bundle can depend on the string structure.

## [edit] 3 Further discussion

The map has homotopy fiber and is the pullback of the path-loop fibration

In the case , it follows from this result that the Pontryagin class maps to a certain multiple (see here for the value of ). In these cases, the generator is usally denoted by , although itself is indivisible.

In the cases the class equals respectively .

By obstruction theory, it follows that a map lifts to if and only if the "characteristic" class

In particular, since the Stiefel-Whitney classes of a vector bundle are independent of an orientation, this answers the question for .

For every -structure on a vector bundle there exists an opposite -structure (inducing the opposite orientation) defined as follows: The connective cover construction is functorial, thus the non-trivial deck transformation of the -fold cover induces a self-map of . Composing the -structure with this self-map of gives the opposite structure. We have since is a self-equivalence which for is the identity on the Pontryagin classes. Thus for a -structure and its opposite, either both lift to a -structure or both do not.

In general, we are given two maps (i.e. two stable vector bundles over with -structure) for which we assume that the compositions with are homotopic (i.e. the bundles are isomorphic as bundles with -structure). We have to investigate whether it is possible that . Here let us assume that and are consecutive connective covers in the sense that

Since the compositions of and with are homotopic, it follows that and differ by a map from to the homotopy fiber of . More precisely, the map is a map of H-spaces, and given as above, there exists a map such that is homotopic to the composition

where the last map is the -space multiplication.

Under the -space multiplication pulls back to

Now it follows that .

Now we choose and as the "universal" example; thus we have to know whether .

For we need to know the pullback of under . This is zero, since the reduction modulo is trivial, as the reduction of modulo is the image of . See [Stong1963, p.539].

For higher we show that by considering reductions modulo , which have been computed for by [Stong1963] and for odd by [Giambalvo1969].

For we need to know the pullback of under . This is a non-zero class: it suffices to show that its reduction modulo is nontrivial. This follows from [Giambalvo1969, Theorem 1']. (The reduction modulo of is , so that the reduction modulo of is zero.)

Thus for example the trivial stable vector bundle on admits a string structure which does not lift to a -structure.

In all higher dimensions , the (reductions modulo of) are non-zero by [Stong1963, p.539].

## [edit] 4 References

- [Giambalvo1969] V. Giambalvo,
*The cohomology of*, Proc. Amer. Math. Soc.**20**(1969), 593–597. MR0236913 (38 #5206) Zbl 0176.52601 - [Stong1963] R. E. Stong,
*Determination of and*, Trans. Amer. Math. Soc.**107**(1963), 526–544. MR0151963 (27 #1944) Zbl 0116.14702 - [Čadek&Crabb&Vanvzura2008] M. Čadek, M. Crabb and J. Vanžura,
*Obstruction theory on 8-manifolds*, Manuscripta Math.**127**(2008), no.2, 167–186. MR2442894 (2009f:55015) Zbl 1157.55011