Sphere bundles and spin (Ex)
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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 $S^2$. | ||
# 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 $S^3$. | # 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. | + | # 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!). |
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− | 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> | </wikitex> | ||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
[[Category:Exercises]] | [[Category:Exercises]] |
Revision as of 22:45, 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!).