Tangent bundles of bundles (Ex)

Latest revision as of 22:28, 29 May 2012

Let $<wikitex>; Let $F \to E \stackrel{\pi}{\to} B$ be a smooth fiber bundle so that all spaces $F, E$ and $B$ are manifolds. {{beginthm|Exercise}} Show that $TE$, the tangent bundle of $E$, splits as the sum of two bundles $$ TE \cong \pi^*TB \oplus T_{\pi}E $$ where $T_{\pi}E$ consists of those tangent vectors tangent to the fibres of $\pi$. {{endthm}} {{beginthm|Question}} Is the bundle $T_{\pi}E$ the pullback of some bundle over $B$? {{endthm}} {{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. {{endthm}} {{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 $TB$. {{endthm}} 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 Pontrjagin class of $\Hh P^n$, [[Wikipedia:Quaternionic_projective_space|quaternionic projective space]]. (This was first achieved in {{citeD|Hirzebruch1953}}). </wikitex> == References == {{#RefList:}} [[Category:Exercises]]F \to E \stackrel{\pi}{\to} B$ be a smooth fiber bundle so that all spaces $F, E$ and $B$ are manifolds.

Exercise 0.1. Show that $TE$, the tangent bundle of $E$, splits as the sum of two bundles

$$\displaystyle TE \cong \pi^*TB \oplus T_{\pi}E$$

where $T_{\pi}E$ consists of those tangent vectors tangent to the fibres of $\pi$.

As an explicit example, recall that there are principal bundles $S^1 \to S^{4k+3} \to \CP^{2k+1}$ and $S^3 \to S^{4k+3} \to \Hh P^k$. 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 $S^3 \to S^3/S^1 \cong S^2$.

Exercise 0.5. Compute the total Pontrjagin class of $\Hh P^n$, quaternionic projective space. (This was first achieved in [Hirzebruch1953]).