Sphere bundles and spin (Ex)

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

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   $M$ a smooth closed manifold, let $S^1\hookrightarrow M^m$$S^1\hookrightarrow M^m$ be a nullhomotopic embedding. For

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   $M$ spin, show that there is more than one possible effect for a surgery on this embedding. Now suppose

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   $M$ is simply connected. For

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   $M$ not spin, show that the effect of a surgery on this embedding is uniquely determined (difficult!).