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==== The tautological and Hopf bundles
Projective space may also be viewed as the quotient under
a group action. Consider a space $Y$ and a group $G$. An
action of $G$ on $Y$ is a map $G\times Y \map Y$,
written $(g,y) \mapsto g.y$, that satisfies $e.y = y$ for all
$y$, where $e$ is the neutral element, and $g.(h.y) = (gh).y$
for all $g,h \in G$, $y \in Y$. The _orbit_ of point $y \in Y$
under the action of $G$ is the set $G.y = \sett{g.y}{ g \in G}$.
The quotient of $Y$ by $G$, written $Y/G$, is the set of orbits.
Note that $Y/G$ is just the set of equivalence classes of elements
of $Y$ under the relation $y' \sim y$ if there is an element $g \in G$
such that $y' = g.y$.
Returning to the construction of projective space,
the map
$(\lambda, Z) \mapsto \lambda Z$
defines an
action of the group $\CC^*$ on
$\CC^{n+1} - \set{0}$, and
\[
\CP^n = (\CC^{n+1} - \set{0})\big/\CC^*
\]
is the quotient. If $p: \CC^{n+1} - \set{0} \map \CP^n$
is the projection map, then the set
\[
p^{-1}([Z]) = \set{ Z \in [Z]} \cong \CC^*,
\]
the so-called
_fiber_ of $p$ at $[Z]$ can be identified with $\CC^*$.
This leads to the important diagram
\[
\CC^* \map \CC^{n+1} - \set{0} \mapright{p} \CP^n,
\]
which defines a $\CC^*$ bundle over $\CP^n$ called the
_tautological bundle_.
It plays a key role in the theory of projective varieties.
FIG
Consider now the unit sphere
\[
S^{2n+1} = \sett{ Z \in \CC^{n+1} } { ||Z|| = 1 }.
\]
It is stable under the action of unit circle subgroup
\[
U = \sett{ \lambda \in \CC}{ |\lambda| = 1 }.
\]
The natural map
\[
S^{2n+1} \map \CC^{n+1} - \set{0}
\]
restricts ot a map of quotient spaces
\[
S^{2n+1}/U \map (\CC^{n+1} - \set{0})/\CC^*,
\]
and it is easy to see that this map is an isomorphism.
This gives a second presentation of projective space
as the quotient of a sphere, and it leads to the famous
_Hopf fibration_,
\[
S^1 \map S^{2n+1} \mapright{h} \CP^n.
\]
The Hopf map is, by the way, the generator of the homotopy
group $\pi_3(S^2)$.
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