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

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[edit] 1 Question

Let E\to X be a (stable) vector bundle. This has a classifying map X\to BO.

A BO\langle n\rangle-structure on E is (the vertical homotopy class of) a lift of the classifying map to a map X\to BO\langle n \rangle. (For n=2,4,8 this is an orientation, a spin structure, a string structure respectively.)

Since for n>m, the map BO\langle n \rangle \to BO factors through BO\langle m \rangle \to BO, a BO\langle n\rangle-structure induces a BO\langle m\rangle-structure, or, vice versa, this specific BO\langle m\rangle-structure can be lifted to a BO\langle n\rangle-structure.

Question 1.1. Given a vector bundle X\to BO and two BO\langle n\rangle-structures on it, is it possible that one of the BO\langle n\rangle-structures can be lifted to a BO\langle n+1\rangle-structure and the other can't?

[edit] 2 Answers

This is not possible for n=2 and n=4, 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 \frac{p_1}{2} does not depend upon the choice of spin structure see [Čadek&Crabb&Vanvzura2008, Defintion p.170].

The answer is however yes for all larger n with BO\langle n\rangle\ne BO\langle n+1\rangle. For example the existence of a BO\langle 9\rangle-structure on a string vector bundle can depend on the string structure.

[edit] 3 Further discussion

The map BO\langle n+1 \rangle \to BO\langle n \rangle has homotopy fiber K(\pi_{n}BO,n-1) and is the pullback of the path-loop fibration

\displaystyle \Omega K(\pi_{n}BO,n) \to PK(\pi_{n}BO,n) \to K(\pi_{n}BO,n)
via a map BO\langle n\rangle \to  K(\pi_{n}BO,n), corresponding to a class k\in H^{n}(BO\langle n\rangle;\pi_{n}BO). Note that this is the generator of H^{n}(BO\langle n\rangle;\pi_{n}BO)\cong Hom(\pi_{n}BO;\pi_{n}BO).

In the case n=4i, it follows from this result that the Pontryagin class p_i\in H^{4i}(BO;\Zz) maps to a certain multiple b_i\cdot k\in H^{4i}(BO\langle 4i\rangle;\Zz) (see here for the value of b_i). In these cases, the generator k is usally denoted by \frac{p_i}{b_i}, although p_i\in H^{4i}(BO;\Zz) itself is indivisible.

In the cases n=1,2 the class k equals w_1\in H^1(BO;\Zz_2) respectively w_2\in H^2(BSO;\Zz_2).

By obstruction theory, it follows that a map f:X\to BO\langle n \rangle lifts to BO\langle n+1\rangle if and only if the "characteristic" class

\displaystyle f^*k\in  H^{n}(X;\pi_{n}BO)

In particular, since the Stiefel-Whitney classes of a vector bundle are independent of an orientation, this answers the question for n=2.

For every BO\langle n\rangle-structure on a vector bundle X\to BO there exists an opposite BO\langle n\rangle-structure (inducing the opposite orientation) defined as follows: The connective cover construction is functorial, thus the non-trivial deck transformation \tau of the 2-fold cover BSO\to BO induces a self-map \tau of BO\langle n\rangle. Composing the BO\langle n\rangle-structure with this self-map of BO\langle n\rangle gives the opposite structure. We have \tau^*(k)=k since \tau is a self-equivalence which for n=4k is the identity on the Pontryagin classes. Thus for a BO\langle n\rangle-structure and its opposite, either both lift to a BO\langle n+1\rangle-structure or both do not.

In general, we are given two maps f,g:X\to BO\langle n \rangle (i.e. two stable vector bundles over X with BO\langle n \rangle-structure) for which we assume that the compositions with BO\langle n \rangle \to BO\langle m \rangle are homotopic (i.e. the bundles are isomorphic as bundles with BO\langle m \rangle-structure). We have to investigate whether it is possible that f^*k\ne g^*k=0. Here let us assume that BO\langle n\rangle and BO\langle m\rangle are consecutive connective covers in the sense that

\displaystyle BO\langle n\rangle = BO\langle n-1\rangle = \dots = BO\langle m+1\rangle \ne BO\langle m\rangle.

Since the compositions of f and g with BO\langle n \rangle \to BO\langle m \rangle are homotopic, it follows that f and g differ by a map from X to the homotopy fiber K(\pi_{m}BO,m-1) of BO\langle n \rangle \to BO\langle m \rangle. More precisely, the map BO\langle n \rangle \to BO\langle m \rangle is a map of H-spaces, and given f,g as above, there exists a map h:X\to K(\pi_{m}BO,m-1) such that f is homotopic to the composition

\displaystyle  X \stackrel{g,h}{\longrightarrow} BO\langle n  \rangle\times K(\pi_{m}BO,m-1) \stackrel{id\times i}\longrightarrow BO\langle n  \rangle\times BO\langle n  \rangle\to BO\langle n \rangle

where the last map is the H-space multiplication.

Under the H-space multiplication k pulls back to

\displaystyle k\otimes 1 + 1\otimes k \in H^*(BO\langle n\rangle)\otimes H^*(BO\langle n\rangle)\subseteq H^*(BO\langle n\rangle \times BO\langle n\rangle).

Now it follows that f^*k=(g,ih)^*(k\otimes 1 + 1\otimes k)=g^*k+h^*i^*k.

Now we choose X=K(\pi_{m}BO,m-1) and h=id as the "universal" example; thus we have to know whether i^*(k)=0.

For n=4,m=2 we need to know the pullback of k=\frac{p_1}{2} under i: K(\Zz_2,1)\to BSpin. This is zero, since the reduction modulo 2 is trivial, as the reduction of k modulo 2 is the image of w_4\in H^4(BO;\Zz_2). See [Stong1963, p.539].

For higher n,m we show that i^*(k)\ne 0 by considering reductions modulo p, which have been computed for p=2 by [Stong1963] and for p odd by [Giambalvo1969].

For n=8,m=4 we need to know the pullback of k=\frac{p_2}{6} under i: K(\Zz,3)\to BString. This is a non-zero class: it suffices to show that its reduction modulo 3 is nontrivial. This follows from [Giambalvo1969, Theorem 1']. (The reduction modulo 2 of k is w_8, so that the reduction modulo 2 of i^*k is zero.)

Thus for example the trivial stable vector bundle on K(\Zz,3) admits a string structure which does not lift to a BO\langle 9\rangle-structure.

In all higher dimensions n, the (reductions modulo 2 of) i^*k are non-zero by [Stong1963, p.539].

[edit] 4 References

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