Sphere bundles and spin (Ex)

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  • For k\geq2, determine the number of distinct linear S^k-bundles over S^2.
  • Let E_k\to S^2 be the complex plane bundle with Euler number k. Explain how to obtain the total space of the sphere-bundle S(E_k) via surgery on S^3.

For m\geq 4 and M a smooth closed manifold, let S^1\hookrightarrow M^m be a nullhomotopic embedding. For M spin, show that there is more than one possible diffeormorphism type for the outcome of a surgery on this embedding.

Now suppose M is simply connected. For M 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

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