Sphere bundles and spin (Ex)

Revision as of 22:44, 19 March 2012

1. For $<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 $m\geq 4$, 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 $S^k$-bundles over $S^2$.
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 $S^3$.
3. For $m\geq 4$, 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!).

References