Does the existence of a string structure depend on a spin structure ? (and a generalization)
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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?
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 answer is however yes for larger . For example the existence of a -structure on a string vector bundle can depend on the string structure.
3 Further discussion
The map has homotopy fiber and is the pullback of the path-loop fibration
By obstruction theory, it follows that a map lifts to if and only if the "characteristic" class
This is for the first and second Stiefel-Whitney class of the corresponding vector bundle over . In particular, since the Stiefel-Whitney classes of a vector bundle are independent of an orientation, this answers the question for .
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.
For we need to know the pullback of under . This is a non-zero class: it suffices to show that its reduction modulo 3 is nontrivial. This follows from [Giambalvo1969, Theorem 1'].
Thus for example the trivial stable vector bundle on admits a string structure which does not lift to a -structure.
The situation is similar in higher dimensions divisible by 4.
4 References
- [Giambalvo1969] V. Giambalvo, The cohomology of , Proc. Amer. Math. Soc. 20 (1969), 593–597. MR0236913 (38 #5206) Zbl 0176.52601