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 $S^3$.
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* 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 effect for a surgery on this embedding.
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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 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 outcome of a surgery on this embedding ''is'' uniquely determined up to diffeomorphism (difficult!).
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== References ==
== References ==
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{{#RefList:}}
[[Category:Exercises]]
[[Category:Exercises]]

Revision as of 12:06, 23 March 2012

  • 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 total space of the sphere-bundle S(E_k) via surgery on S^3.
For m\geq 4 and
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a smooth closed manifold, let S^1\hookrightarrow M^m be a nullhomotopic embedding. For
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spin, show that a priori there is more than one possible diffeormorphism type for the outcome of a surgery on this embedding. Now suppose
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is simply connected. For
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not spin, show that the outcome of a surgery on this embedding is uniquely determined up to diffeomorphism (difficult!).

References

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