Tangent bundles of bundles (Ex)
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Suppose that $\pi \colon E \to B$ is the sphere bundle of a vector bundle. Determine the stable tangent bundle of $E$ in terms of $\pi$ and $TB$. | Suppose that $\pi \colon E \to B$ is the sphere bundle of a vector bundle. Determine the stable tangent bundle of $E$ in terms of $\pi$ and $TB$. | ||
{{endthm}} | {{endthm}} | ||
+ | {{beginthm|Exercise}} | ||
+ | Compute the total Pontragin class of $\Hh P^n$, [[Wikipedia:Quaternionic_projective_space|quaternionicprojective space]]. (This was first achieved in {{citeD|Hirzebruch1953}}). | ||
</wikitex> | </wikitex> | ||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
[[Category:Exercises]] | [[Category:Exercises]] |
Revision as of 18:02, 9 February 2012
Let be a smooth fiber bundle so that all spaces and are manifolds.
Exercise 0.1. Show that , the tangent bundle of , splits as the sum of two bundles
where consists of those tangent vectors tangent to the fibres of .
Exercise 0.2. Suppose that is itself a smooth vector bundle. Determine in terms of and regarded as a vector bundle.
Exercise 0.3. Suppose that is the sphere bundle of a vector bundle. Determine the stable tangent bundle of in terms of and .
Exercise 0.4.
Compute the total Pontragin class of , quaternionicprojective space. (This was first achieved in [Hirzebruch1953]).