Sphere bundles and spin (Ex)

• For $k\geq2$$<wikitex>; * For $k\geq2$, determine the number of distinct linear $S^k$-bundles over $S^2$. * Let $E_k\to S^2$ be the complex plane bundle with Euler number $k$. Explain how to obtain the total space of the sphere-bundle $S(E_k)$ via surgery on $S^3$. For $m\geq 4$ and $M$ a smooth closed manifold, let $S^1\hookrightarrow M^m$ be a nullhomotopic embedding. For $M$ spin, show that ''a priori'' there is more than one possible diffeormorphism type for the outcome of a surgery on this embedding. Now suppose $M$ is simply connected. For $M$ not spin, show that the outcome of a surgery on this embedding ''is'' uniquely determined up to diffeomorphism (difficult!). </wikitex> == References == {{#RefList:}} [[Category:Exercises]]k\geq2$, determine the number of distinct linear $S^k$$S^k$-bundles over $S^2$$S^2$.
• Let $E_k\to S^2$$E_k\to S^2$ be the complex plane bundle with Euler number $k$$k$. Explain how to obtain the total space of the sphere-bundle $S(E_k)$$S(E_k)$ via surgery on $S^3$$S^3$.

For $m\geq 4$$m\geq 4$ and $M$$M$ a smooth closed manifold, let $S^1\hookrightarrow M^m$$S^1\hookrightarrow M^m$ be a nullhomotopic embedding. For $M$$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$$M$ is simply connected. For $M$$M$ not spin, show that the outcome of a surgery on this embedding is uniquely determined up to diffeomorphism (difficult!).