Sphere bundles and spin (Ex)
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* For $k\geq2$, determine the number of distinct linear $S^k$-bundles over $S^2$. | * 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 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 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 | + | 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 | + | 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!). |
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== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
[[Category:Exercises]] | [[Category:Exercises]] |
Revision as of 12:06, 23 March 2012
- For , determine the number of distinct linear -bundles over .
- Let be the complex plane bundle with Euler number . Explain how to obtain the total space of the sphere-bundle via surgery on .
Tex syntax errora smooth closed manifold, let be a nullhomotopic embedding. For
Tex syntax errorspin, show that a priori there is more than one possible diffeormorphism type for the outcome of a surgery on this embedding. Now suppose
Tex syntax erroris simply connected. For
Tex syntax errornot spin, show that the outcome of a surgery on this embedding is uniquely determined up to diffeomorphism (difficult!).