Sphere bundles and spin (Ex)
(Difference between revisions)
Patrickorson (Talk | contribs) |
Patrickorson (Talk | contribs) |
||
Line 1: | Line 1: | ||
<wikitex>; | <wikitex>; | ||
# 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 | + | |
− | 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. | ||
Revision as of 22:44, 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!).