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
![\displaystyle \Omega K(\pi_{n}BO,n) \to PK(\pi_{n}BO,n) \to K(\pi_{n}BO,n)](/images/math/2/4/3/2432df5b8ead31b3bfabdc13c08ee7a8.png)
![BO\langle n\rangle \to K(\pi_{n}BO,n)](/images/math/5/0/7/507f5afde7f0c4ab2c23c108e435a749.png)
![k\in H^{n}(BO\langle n\rangle;\pi_{n}BO)](/images/math/2/2/6/22655c2d9f86060eec6424fb3b396700.png)
![H^{n}(BO\langle n\rangle;\pi_{n}BO)\cong Hom(\pi_{n}BO;\pi_{n}BO)](/images/math/2/5/5/255f24b1f71a5ef819099f15194f753b.png)
By obstruction theory, it follows that a map lifts to
if and only if the "characteristic" class
![\displaystyle f^*k\in H^{n}(X;\pi_{n}BO)](/images/math/9/2/0/9206d61284960cf675e23c333682db22.png)
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
![\displaystyle BO\langle n\rangle = BO\langle n-1\rangle = \dots = BO\langle m+1\rangle \ne BO\langle m\rangle.](/images/math/1/9/7/1970c97853770603cbeb366ab7e1d041.png)
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
![\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](/images/math/5/1/3/513794f749aabf5d6a8da7d512d6b25c.png)
where the last map is the -space multiplication.
Under the -space multiplication
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).](/images/math/6/a/f/6af00f24b2bdb3b3a54456718a4bb4a7.png)
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