Sphere bundles and spin (Ex)
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Patrickorson (Talk | contribs) (Created page with "<wikitex>; For $k\geq2$, determine the number of distinct $S^k$-bundles over $S^2$. For $m\geq 4$, let $S^1\hookrightarrow M^m$ be a nullhomotopic embedding. For $M$ spin, sh...") |
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− | For $k\geq2$, determine the number of distinct $S^k$-bundles over $S^2$. | + | # For $k\geq2$, determine the number of distinct $S^k$-bundles over $S^2$. |
− | 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. | + | # 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!). | Now suppose $M$ is simply connected. For $M$ not spin, show that the effect of a surgery on this embedding ''is'' uniquely determined (difficult!). |
Revision as of 22:43, 19 March 2012
- For , determine the number of distinct -bundles over .
- Let be the complex plane bundle with euler
number . Explain how to obtain the sphere-bundle via surgery on .
- For , let be a nullhomotopic embedding. For spin, show that there is more than one possible effect for a surgery on this embedding.
Now suppose is simply connected. For not spin, show that the effect of a surgery on this embedding is uniquely determined (difficult!).