Tangent bundles of bundles (Ex)
(Difference between revisions)
m |
|||
Line 21: | Line 21: | ||
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 $S^3 \to S^3/S^1 \cong S^2$. | 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| | + | {{beginthm|Exercise}} Compute the total Pontragin class of $\Hh P^n$, [[Wikipedia:Quaternionic_projective_space|quaternionic projective space]]. (This was first achieved in {{citeD|Hirzebruch1953}}). |
</wikitex> | </wikitex> | ||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
[[Category:Exercises]] | [[Category:Exercises]] |
Revision as of 23:09, 2 March 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 , quaternionic projective space. (This was first achieved in [Hirzebruch1953]).