Sphere bundles and spin (Ex)

Latest revision as of 09:24, 1 April 2012

• For $<wikitex>; * 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 ''a priori'' 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!). </wikitex> == References == {{#RefList:}} [[Category:Exercises]]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.

References

• [Gompf&Stipsicz1999] R. E. Gompf and A. I. Stipsicz, $4$-manifolds and Kirby calculus, American Mathematical Society, 1999. MR1707327 (2000h:57038) Zbl 0933.57020