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}} | ||
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+ | As an explicit example, recall that there are principal bundles $S^1 \to S^{4k+3} \to \CP^{2k+1}$ and $S^3 \to S^{4k+1} \to \Hh P^k$. It follows that there is an $S^2$ fibre bundle $S^2 \to \CP^{2k+1} \to \Hh P^k$ with structure group $S^3$ acting on $S^2$ via $S^3 \to S^3/S^1 \cong S^2$. | ||
{{beginthm|Exercise}} Compute the total Pontragin class of $\Hh P^n$, [[Wikipedia:Quaternionic_projective_space|quaternionicprojective space]]. (This was first achieved in {{citeD|Hirzebruch1953}}). | {{beginthm|Exercise}} 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 18:08, 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 .
Question 0.2. Is the bundle the pullback of some bundle over ?
Exercise 0.3. Suppose that is itself a smooth vector bundle. Determine in terms of and regarded as a vector bundle.
Exercise 0.4. Suppose that is the sphere bundle of a vector bundle. Determine the stable tangent bundle of in terms of and .
As an explicit example, recall that there are principal bundles and . It follows that there is an fibre bundle with structure group acting on via .
Exercise 0.5. Compute the total Pontragin class of , quaternionicprojective space. (This was first achieved in [Hirzebruch1953]).