# Sphere bundles and spin (Ex)

From Manifold Atlas

- For , determine the number of distinct linear -bundles over .
- Let be the complex plane bundle with Euler number . Explain how to obtain the total space of the sphere-bundle via surgery on .

For and a smooth closed manifold, let be a nullhomotopic embedding. For spin, show that there is more than one possible diffeormorphism type for the outcome of a surgery on this embedding.

Now suppose is simply connected. For not spin, show that the outcome of a surgery on this embedding *is* uniquely determined up to diffeomorphism (difficult!).

For hints, see section in [Gompf&Stipsicz1999] on surgery.

## [edit] References

- [Gompf&Stipsicz1999] R. E. Gompf and A. I. Stipsicz,
*-manifolds and Kirby calculus*, American Mathematical Society, 1999. MR1707327 (2000h:57038) Zbl 0933.57020