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\listok{7}{Geometry 7: Smooth fibrations}

{\scriptsize
{\bf Rules:} You may choose to solve only 
``hard'' exercises (marked with !, * and **) 
or ``ordinary'' ones (marked with ! or unmarked),
or both, if you want to have extra problems.
To have a perfect score, a student must obtain
(in average) a score of 10 points per week.
It's up to you to ignore handouts entirely,
because passing tests in class and having
good scores at final exams could compensate
(at least, partially) for the points obtained 
by grading handouts.

Solutions for the problems are to be explained to the 
examiners orally in the class and marked in the score sheet. 
It's better to have a written version of your solution with 
you. It's OK to share your solutions with other students, and use
books, Google search and Wikipedia, we encourage it.

If you have got credit for 2/3 of ordinary problems
or 2/3 of ``hard'' problems, you receive  
$6t$ points, where $t$ is a number depending on the
date when it is done. Passing all ``hard'' 
or all ``ordinary'' problems
(except at most 2) brings you $10t$ points.
Solving of ``**'' (extra hard) problems is not
obligatory, but each such problem gives you a credit
for 2 ``*'' or ``!'' problems in the ``hard'' set.

The first 3 weeks after giving a handout, $t=1.5$,
between 21 and 35 days, $t=1$, and afterwards, $t=0.7$.
The scores are not cumulative, only the
best score for each handout counts.

Please keep your score sheets until the final
evaluation is given.
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Locally trivial smooth fibrations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $M \stackrel \phi \arrow N$ be a differentiable
map of smooth manifolds. A {\bf critical point} of $\phi$ is a point
$m\in M$ where its differential has rank less than maximal
possible: $r<\min(\dim M, \dim N)$. 
\ed

\exercise
Let $M \stackrel \phi \arrow N$ be a map without critical points, 
$\dim M > \dim N$, and $X\subset N$ a smooth submanifold. Prove that
 $\phi^{-1}(X)$  is a smooth submanifold in $M$.
\ez

\hint Use the implicit function theorem.
\eh

\definition
{\bf A trivial smooth fibration} is a projection
$N \times U \arrow U$, where $N$ and $U$ are smooth manifolds.
\ed

\definition
A surjective smooth map of manifolds
$M \stackrel \phi \arrow N$  is called
{\bf a locally trivial smooth fibration} if
each $x\in N$ has a neighbourhood $U\ni x$
such that the projection $\phi^{-1}(U)\arrow U$ 
is a trivial smooth fibration.
\ed

\remark
Let $M \stackrel \phi \arrow N$ 
be a locally trivial smooth fibration, and
$U\subset N$ an open subset. The map
$\phi^{-1}(U)\arrow U$ is called {\bf restriction
of the locally trivial fibration to $U\subset N$.}
\er

\exercise
Show that any locally trivial fibration is a map
without critical points.
\ez

\exercise
Consider a 3-dimensional sphere $S^3\subset \R^4=\C^2$, 
and let $\pi:\; S^3\arrow \C P^1$ be a projection induced
by the tautological map $\C^2\backslash 0 \arrow \C P^1$.
Show that it is a locally trivial fibration with fiber $S^1$.
\ez

\remark
This map is called {\bf the Hopf fibration}.
\er

\exercise
Let $\pi:\; S^3\arrow \C P^1$ be a Hopf fibration
and $\C=\C P^1\backslash \{0\}\hookrightarrow \C P^1$ 
the standard embedding.
Prove that $\pi^{-1}(\C P^1\backslash \{0\})$ 
is homeomorphic to $S^1 \times \R^2$.
\ez

\exercise
Prove that Hopf fibration is not a trivial fibration.
\ez

\exercise[**]
Prove {\bf Ehresmann theorem}: any surjective,
smooth map of compact manifolds without critical points
is a locally trivial fibration.
\ez

\exercise
\enum
\ite 
Construct a surjective map $S^{2n+1}\arrow \C P^n$
without critical points. 
\ite[**] Prove that this is a
locally trivial, but non-trivial fibration.
\ee
\ez

\hint
Generalize the construction of Hopf fibration.
\eh

\exercise[**]
Construct a locally trivial smooth fibration
$S^7\arrow S^4$. Prove that it is non-trivial.
\ez

\definition
Let $M_1 \stackrel {\pi_1} \arrow N$ and
$M_2\stackrel {\pi_2}\arrow N$ be continuous maps
of topological spaces, and $\Delta \subset
N \times N$ a diagonal. Let
$\pi_1\times \pi_2:\; M_1\times M_2 \arrow N \times N$
be a natural projection. Define
$M_1\times_N M_2:=(\pi_1\times \pi_2)^{-1}(\Delta)$.
The space $M_1\times_N M_2$
is called {\bf a fibered product}, or {\bf fiber product 
of $M_1$ and $M_2$
over $N$.}
\ed


\exercise[!]
Let $M_1 \stackrel {\pi_1} \arrow N$ and
$N_2\stackrel {\pi_2}\arrow N$ be locally trivial 
smooth fibrations with fibers $F_1$ and $F_2$.
Prove that the natural map
$M_1\times_N M_2\arrow N$
is a locally trivial fibration with fiber
$F_1\times F_2$.
\ez


\exercise
Represent a Moebius strip as a smooth fibration
 $М \stackrel \pi\arrow S^1$ with fiber $]0,1[$.
Prove that $M \times_{S^1} M$ is homeomorphic to
$S^1\times ]0,1[ \times ]0,1[$.
\ez

\exercise[*]
Let $\pi:\; S^3\arrow \C P^1$ be a Hopf fibration.
Prove that $S^3\times_{\C P^1} S^3$ is homeomorphic to
$S^3 \times S^1$.
\ez
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Groups and fiber products}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
{\bf A topological group} is a topological space
equipped with the group operations (product and
taking inverse) which are continuous and satisfy the group
axioms.
\ed

\exercise
Let $G$ be a subgroup of the group of matrices, with natural topology.
Prove that it is a topological group.
\ez

\exercise
Construct a structure of topological group on $S^3$.
\ez

\exercise[*]
Can an even-dimensional sphere be a topological group?
\ez

\exercise[*]
Can a bouquet of two circles be a topological group?
\ez

\definition
Let  $M  \stackrel f \arrow N$, $M' \stackrel {f'}\arrow N$ 
be continuous maps (morphisms) of topological spaces.
A morphism $M \stackrel \psi \arrow M'$ is called {\bf a 
morphism over $N$}, if the following diagram is commutative:
\[ \begin{CD}
M @>{\psi}>>  M'\\
@V{f}VV @V{f'}VV\\
N @>{\Id}>> N
\end{CD}
\]
\ed

%\exercise
%Let $B\stackrel \pi \arrow M$
%be a continuous map, and $B\stackrel \Delta \arrow B\times_M B$
%maps $b\in B$ to $(b,b)\in B\times_M B$.
%Show that this is a morphism over $B$.
%\ez


\definition
Let $B \stackrel \pi \arrow M$ be a continuous map, 
and $B\times_M B \stackrel \Psi \arrow M$ -
a morphism over $M$. This morphism is called {\bf
associative multiplication} if it is associative on the fibers of
 $\pi$, that is, satisfies
$\Psi(a, \Psi(b,c))= \Psi(\Psi(a,b),c)$ for every
triple $a,b,c$ in the same fiber.
A section $M \stackrel e \arrow B$ is called
{\bf the unit} if the maps
\[ B \stackrel {\Id_B \times e}\arrow B\times_M B \stackrel \Psi \arrow B\]
and \[ B \stackrel {e\times \Id_B}\arrow B\times_M B \stackrel \Psi \arrow B\]
are equal to $Id_B$.  A morphism $\nu:\; B \arrow B$ over $M$ 
is called {\bf group inverse} if each of the maps
\[ B\stackrel \Delta \arrow B\times_M B \stackrel
{\Id_B\times \nu}\arrow B\times_M B \stackrel \Psi \arrow B\]
and \[ B\stackrel \Delta \arrow B\times_M B \stackrel
{\nu\times \Id_B}\arrow B\times_M B \stackrel \Psi \arrow B\]
 is a constant map, mapping $b$ to $e(\pi(b))$. A map $B \stackrel \pi \arrow M$
equipped with associative multiplication, unit and 
group inverse is called {\bf a topological group over $M$}. 
\ed

\exercise
Let $B \stackrel \pi \arrow M$ 
be a topological group over $M$. 
Show that all fibers of $\pi$ ara topological groups.
\ez

\exercise
Let $G \times M \arrow M$ be a trivial fibration.
Assume that $G$ is equipped with a set of continuous
group operations, indexed by $m\in M$ and continuously
depending on $m$ (that is, the corresponding maps, say,
$G\times G \times M \arrow G$ are continuous).
Prove that this date gives a structure of topological
group over $M$ on $G\times M$.
\ez

\exercise
Let $B$ be a topological group over $M$.
Consider the space of continuous sections
$M \arrow B$. Prove that it is a group.
\ez


%\exercise[*]
%В условиях предыдущей задачи, рассмотрим
%открыто-компактную топологию на пространстве
%непрерывных сечений $M \arrow B$. Докажите, что
%пространство сечений будет топологической
%группой.
%\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Vector bundles and smooth fibrations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
\label{_vector_spaces_Exercise_}
Let $G$ be an abelian group, and $k$ a field.
Suppose that for each non-zero $\lambda\in k$
there exists an automorphism $\phi_\lambda:\; G \arrow G$,
such that  $\phi_\lambda\circ\phi_{\lambda'}=\phi_{\lambda\lambda'}$,
and $\phi_{\lambda+\lambda'}(g)= \phi_{\lambda}(g)+\phi_{\lambda'}(g).$
Show that $G$ is a vector space over $k$. Show that all
vector spaces can be obtained this way.
\ez

\definition
Let $k=\R$ or $\C$.
An abelian topological group $B\stackrel \pi \arrow M$ over $M$
is called {\bf relative vector space over $M$}
if for each non-zero $\lambda\in k$ there exists a
continuous automorphism
$\phi_\lambda:\; B \arrow B$ of a group $B$ over $M$
satisfying assumptions of Exercise \ref{_vector_spaces_Exercise_},
such that the corresponding map $B\times k \arrow B$ is continuous.
\ed



\exercise
Let $B\stackrel \pi \arrow M$ be a relative vector space
over $M$, $U\subset M$ an open subset,
and ${\cal B}(U)$ the space of sections of a map 
$\pi^{-1}(U) \stackrel \pi \arrow U$.
\enum
\ite Show that
 ${\cal B}(U)$ is a vector space.
\ite Prove that ${\cal B}(U)$ defines a sheaf of modules
over a sheaf $C^0(M)$ of continuous functions.
\ee
\ez



\exercise
Let  $S\subset \R^n$ be a subset (not necessarily 
a smooth submanifold), $s\in S$ a point, and $v\in
T_s\R^n$ a vector. We sat that $v$ belongs to a {\bf tangent cone} 
$C_sS$ if the distance from $S$ to a point $s+tv$
converges to 0 as $t\rightarrow 0$ faster than linearly:
\[
\lim_{t \arrow 0}\frac{d(S, s+tv)}{t} \arrow 0.
\]
\enum
\ite[!]
Let $T_sS$ be a space generated by $C_s S$.
Show that the set $TS$ of all pairs $(s, v), s\in S, v\in T_s S$
is a relative vector space over $S$.\footnote{The space $CS$ is called {\bf 
a tangent cone to к $S$}.}

\ite[!] 
Find $CS$ for  set $S\subset \R^3$ of zeros 
of a polynomial $x^2+y^2-z^2$.

\ite[!] Show that in this situation,
$CS\arrow S$ is not a locally trivial smooth fibration.
\ee
\ez

\definition
Let $B \arrow M$ be a smooth locally trivial
fibration with fiber $\R^n$.
Assume that  $B$ is equipped with a structure
of relative vector space over $M$, and all the maps
used in the definition of a relative vector space
are smooth. Then $B$
is called {\bf a total space of a vector bundle.}
\ed

\exercise
Let $B \arrow M$ be a relative vector space over $M$,
and ${\cal F}$ the corresponding sheaf of sections.
Prove that it is a locally free sheaf of 
$C^\infty M$-modules.
\ez

\definition
Recall that {\bf a vector bundle}
is a locally free sheaf of modules over
 $C^\infty M$. A vector bundle is called
{\bf trivial} if it is isomorphic to $C^\infty M^n$.
\ed

\definition
Let ${\cal B}$ be an $n$-dimensional vector bundle on $M$,
$x\in M$ a point,  ${\cal B}_x$ the space of germs of ${\cal B}$
in $x$, and ${\goth m}_x\subset C^\infty_x M$
the maximal ideal in the ring of germs $C^\infty_x M$
of smooth functions. Define {\bf the fiber} of
${\cal B}$ in $x$ as a quotient
${\cal B}_x/{\goth m}_x{\cal B}_x$.
A fiber of a vector bundle is denoted
${\cal B}\restrict x$.
\ed

\exercise
Show that a fiber of an $n$-dimensional bundle
is an $n$-dimensional vector space.
\ez

\exercise
\label{_bije_fiber_Exercise_}
Let  ${\cal B}= C^\infty M^n$ be a trivial $n$-dimensional
bundle on $M$, and  
$b\in {\cal B}\restrict x$ a point of a fiber, represented
by a germ $\phi \in {\cal B}_x=C^\infty_m M^n$,
$\phi=(f_1, ..., f_n)$. Consider a map from the set
of all fibers ${\cal B}$ to $M \times \R^n$,
mapping $(x, \phi=(f_1, ..., f_n))$
to $(f_1(x), ..., f_n(x))$. Prove that this map is bijective.
\ez

\definition
Let ${\cal B}$ be an $n$-dimensional vector bundle 
over $M$. Denote the set of all
vectors in all fibers of  ${\cal B}$ over all points of $M$
by $\Tot {\cal B}$. Let  $U\subset M$ be an open subset of $M$,
with  ${\cal B}\restrict U$ a trivial bundle.
Using the local bijection $\Tot {\cal B}(U)=U \times \R^n$
defined in Exercise \ref{_bije_fiber_Exercise_},
we consider topology on $\Tot {\cal B}$
induced by open subsets in 
$\Tot {\cal B}(U)=U \times \R^n$ for all
open subsets $U\subset M$ and all trivializations of
${\cal B}\restrict U$. 
\ed

\exercise
Show that  $\Tot {\cal B}$ with this topology is
a locally trivial fibration over $M$, with fiber $\R^n$.
\ez

\exercise[!]
Show that $\Tot {\cal B}$
is equipped with a natural structure
of a relative vector space over $M$, and
the sheaf of smooth sections of $\Tot {\cal B}\arrow M$
is isomorphic to ${\cal B}$.
\ez

\definition
Let ${\cal B}$ be a vector bundle on $M$.
Then $B=\Tot {\cal B}$ is called {\bf the total space 
of a vector bundle ${\cal B}$}.
\ed

\remark
In practice, ``the total space of a vector bundle''
is usually denoted by the same letter as the 
corresponding sheaf. Quite often, mathematicians
don't even distinguish between these two notions.
\er

\exercise
Let $M_1\stackrel \phi \arrow M$ be a smooth
map of manifolds, and 
$B \stackrel \pi \arrow M$ a total space of a vector bundle. 
Prove that $B \times_M M_1$
is a total space of a vector bundle on
$M_1$. 
\ez

\definition
This bundle is denoted $\phi^* B$, and called
{\bf inverse image}, or {\bf a pullback}
of $B$.
\ed

\exercise
Prove that the fiber $\phi^*(B)\restrict x$
is naturally identified with $B\restrict {\phi(x)}$.
\ez

\exercise
Prove that a pullback of a trivial bundle is trivial.
\ez

\exercise
Let $M_1\stackrel \phi \arrow M$ be a surjective, smooth
map without critical points, and  $B$ a non-trivial
bundle on $M$. 
\enum
\ite[*] Can the bundle $\phi^* B$ be trivial?
\ite[*] Suppose that $M_1$ is compact.  Can $\phi^* B$ be trivial?
\ee
\ez




\end{document}
