Tangent bundles of bundles (Ex)
(Difference between revisions)
m |
m |
||
Line 6: | Line 6: | ||
where $T_{\pi}E$ consists of those tangent vectors tangent to the fibres of $\pi$. | where $T_{\pi}E$ consists of those tangent vectors tangent to the fibres of $\pi$. | ||
{{endthm}} | {{endthm}} | ||
+ | |||
+ | {{beginthem|Question}} | ||
+ | Is the bundle $T_{\pi}E$ the pullback of some bundle over $B$? | ||
+ | {{endthm}} | ||
+ | |||
{{beginthm|Exercise}} | {{beginthm|Exercise}} | ||
Suppose that $\pi \colon E \to B$ is itself a smooth vector bundle. Determine $TE$ in terms of $TB$ and $\pi$ regarded as a vector bundle. | Suppose that $\pi \colon E \to B$ is itself a smooth vector bundle. Determine $TE$ in terms of $TB$ and $\pi$ regarded as a vector bundle. | ||
{{endthm}} | {{endthm}} | ||
+ | |||
{{beginthm|Exercise}} | {{beginthm|Exercise}} | ||
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}} | {{beginthm|Exercise}} | ||
Compute the total Pontragin class of $\Hh P^n$, [[Wikipedia:Quaternionic_projective_space|quaternionicprojective space]]. (This was first achieved in {{citeD|Hirzebruch1953}}). | Compute the total Pontragin class of $\Hh P^n$, [[Wikipedia:Quaternionic_projective_space|quaternionicprojective space]]. (This was first achieved in {{citeD|Hirzebruch1953}}). |
Revision as of 18:03, 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 .
Template:Beginthem Is the bundle the pullback of some bundle over ? </div>
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]).