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%Anan'in's translation of the Russian version
%version 1.1,\ \   11.02.2013, a couple of errors from Pasha Tomas
%version 1.2,\ \   18.02.2013, more minor errors
%version 1.2.1,\ \ 20.02.2013, ** or RP^n to R^{n+1}
%version 1.3,\ \ 25.02.2013, ukazanie dobavil

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\begin{document}

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\listok{1}{Geometry 1: Manifolds and sheaves}

{\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.
The first score sheet will be distributed
February 11-th. 

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.

The original English translation of this handout
was done by Sasha Anan$'$in (UNICAMP) in 2010.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Topological manifolds}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\remark
 Manifolds can be smooth (of a given ``class of
smoothness''), real analytic, or topological (continuous).  
{\bf Topological manifold} is easiest to define, it is
a topological space which is locally homeomorphic to
an open ball in $\R^n$.
\er


\definition
{\bf An action} of a group on a manifold is silently
assumed to be continuous.
Let $G$ be a group acting on a set $M$. The {\bf
stabilizer} of $x\in M$ is the subgroup of all elements in $G$ that
fix $x$. An action is {\bf free} if the stabilizer of every point is
trivial.
\ed

\exercise
Let $G$ be a  finite group acting freely on a
Hausdorff manifold $M$. Show that the quotient space $M/G$ is a
manifold.
\ez

\exercise
Construct an example of a finite group $G$
acting non-freely on a manifold $M$ such that $M/G$ is not a manifold.
\ez

\exercise
Consider the quotient of $\Bbb R^2$ by the action
of $\{\pm1\}$ that maps $x$ to $-x$. Is the quotient space a manifold?
\ez

\exercise[*]
Let $M$ be a path connected, Hausdorff topological manifold, and
$G$ a group of all its homeomorphisms. Prove that
$G$ acts on $M$ transitively.
\ez

\exercise[**]
Prove that any closed subgroup $G\subset GL(n)$
of a matrix group is homeomorphic
to a manifold, or find a counterexample.
\ez


\remark
In the above definition of a manifold,
it is not required to be Hausdorff. 
Nevertheless, in most cases, manifolds are 
silently assumed to be Hausdorff.
\er


\exercise
Construct an example of a non-Hausdorff manifold.
\ez

\exercise
Show that $\Bbb R^2/\Bbb Z^2$ is a manifold.
\ez

\exercise
Let $\alpha$ be an irrational number. The group
$\Bbb Z^2$ acts on $\Bbb R$ by the formula $t\mapsto t+m+n\alpha$.
Show that this action is free, but the quotient $\Bbb R/\Bbb Z^2$ is
not a manifold.
\ez

\exercise[**]
Construct an example of a (non-Hausdorff) manifold of positive
dimension such that the closures of two arbitrary nonempty 
open sets always
intersect, or show that such a manifold does not exist.
\ez

\exercise[**]
 Let $G\subset GL(n,\Bbb R)$ be a compact
subgroup. Show that the quotient space
$GL(n,\Bbb R)/G$ is also a manifold.
\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Smooth manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
 A {\bf cover} of a topological space $X$ is a
family of open sets $\{U_i\}$ such that $\bigcup_iU_i=X$. A cover
$\{V_i\}$ is a {\bf refinement} of a cover $\{U_i\}$ if every $V_i$ is
contained in some $U_i$.
\ed

\exercise
 Show that any two covers of a topological space
admit a common refinement.
\ez

\definition
 A cover $\{U_i\}$ is an {\bf atlas} if for every
$U_i$, we have a map $\varphi_i:U_i\to\R^n$ giving a
homeomorphism of $U_i$ with an open subset in $\R^n$. The {\bf
transition maps}
$$\Phi_{ij}:\varphi_i(U_i\cap U_j)\to\varphi_j(U_i\cap U_j)$$
are induced by the above homeomorphisms. An atlas is {\bf smooth} if
all transition maps are smooth (of~class $C^\infty$, i.e., infinitely
differentiable), {\bf smooth of class} $C^i$ if all transition
functions are of differentiability class $C^i$, 
and~{\bf real analytic} if all transition maps 
admit a Taylor expansion at each point.
\ed

\definition
A {\bf refinement} of an {\bf
atlas} is a refinement of the corresponding cover $V_i\subset U_i$
equipped with the maps $\varphi_i:V_i\to\Bbb R^n$ that are the
restrictions of $\varphi_i:U_i\to\Bbb R^n$. Two atlases
$(U_i,\varphi_i)$ and $(U_i,\psi_i)$ of class $C^\infty$ or $C^i$
(with the same cover) are {\bf equivalent} in this class if, for
all $i$, the map $\psi_i\circ\varphi_i^{-1}$ defined on the
corresponding open subset in $\Bbb R^n$ belongs to the mentioned
class. Two~arbitrary atlases are {\bf equivalent} if the
corresponding covers possess a common refinement giving
equivalent atlases.
\ed

\definition
 A {\bf smooth structure} on a manifold (of class
$C^\infty$ or $C^i$) is an atlas of class $C^\infty$ or $C^i$
considered up to the above equivalence. A {\bf smooth manifold} is a
topological manifold equipped with a smooth structure.
\ed

\remark
Terrible, isn't it?
\er

\exercise[*]
 Construct an example of two nonequivalent
smooth structures on $\Bbb R^n$.
\ez

\definition
A {\bf smooth function} on a manifold
$M$ is a function $f$ whose restriction to the chart $(U_i,\varphi_i)$
gives a smooth function 
$f\circ \varphi_i^{-1}:\; \varphi_i(U_i)\arrow \R$ for each open
subset $\varphi_i(U_i)\subset\Bbb R^n$.
\ed

\remark
 There are several ways to define a smooth manifold. The
above way is most standard. It is not the most convenient one but you
should know it. Two other ways (via sheaves of functions and via
Whitney's theorem) are presented further in these handouts.
\er

\definition
A {\bf presheaf of functions} on a
topological space $M$ is a collection of subrings 
${\cal F}(U)\subset C(U)$ in
the ring $C(U)$ of all functions on $U$, for each open 
subset $U\subset M$,  such that the restriction of
every $\gamma\in{\cal F}(U)$ to an open subset $U_1\subset U$ belongs to
${\cal F}(U_1)$. 
\ed

\definition
A presheaf of functions
${\cal F}$ is called {\bf a sheaf of functions} 
if these subrings
satisfy the following condition. Let $\{U_i\}$ be a cover of an open
subset $U\subset M$ (possibly infinite)
and $f_i\in{\cal F}(U_i)$ a family of
functions defined on the open sets of the cover and 
compatible on the pairwise intersections:
$$f_i|_{U_i\cap U_j}=f_j|_{U_i\cap U_j}$$
for every pair of members of the cover. Then there exists
$f\in{\cal F}(U)$ such that $f_i$ is the restriction of $f$ to $U_i$ for
all $i$.
\ed

\remark
{\bf A presheaf of functions} is a collection of subrings
of functions on open subsets, compatible with
restrictions. {\bf A sheaf of fuctions} is a presheaf
allowing ``gluing'' a function on a bigger open set
if its restriction to smaller open sets lies in the
presheaf.
\er

\definition
A sequence $A_1 \arrow A_2 \arrow A_3 \arrow ...$
of homomorphisms of abelian groups or vector spaces
is called {\bf exact} if the image of each map is
the kernel of the next one.
\ed

\exercise
Let ${\cal F}$ be a presheaf of functions. Show
that ${\cal F}$ is a sheaf if and only if for every cover $\{U_i\}$ of
an open subset $U\subset M$, the sequence of restriction maps
$$0\to{\cal F}(U)\to\prod\limits_i{\cal F}(U_i)\to\prod\limits_{i\ne
j}{\cal F}(U_i\cap U_j)$$
is exact, with $\eta\in {\cal F}(U_i)$ mapped to $\eta\restrict{U_i\cap U_j}$
and $-\eta\restrict{U_j\cap U_i}$. 
\ez

\exercise
Show that the following spaces of functions on
$\Bbb R^n$ define sheaves of functions.
\enum
\ite Space of continuous functions.
\ite Space of smooth functions.
\ite Space of functions of differentiability class $C^i$.
\ite[*] Space of functions that are pointwise limits of sequences of
continuous functions.
\ite Space of functions vanishing outside a set of measure $0$.
\ee
\ez


\exercise
Show that the following spaces of functions on
$\Bbb R^n$ are presheaves, but not sheaves
\enum
\ite Space of constant functions.
\ite Space of bounded functions.
\ite Space of functions vanishing outside of a bounded set.
\ite Space of continuous functions with finite $\int |f|$.
\ee
\ez

\definition
A {\bf ringed space} $(M,{\cal F})$ is a
topological space equipped with a sheaf of functions. A~{\bf morphism}
$(M,{\cal F})\stackrel\Psi\longrightarrow(N,{\cal F}')$ of ringed spaces
is a continuous map $M\stackrel\Psi\longrightarrow N$ such that, for
every open subset $U\subset N$ and every function $f\in{\cal F}'(U)$, the
function $f\circ\Psi$ belongs to the ring
${\cal F}\big(\Psi^{-1}(U)\big)$. An {\bf isomorphism} of ringed spaces
is a homeomorphism $\Psi$ such that $\Psi$ and $\Psi^{-1}$ 
are morphisms of ringed spaces.
\ed

\remark
Usually the term ``ringed space'' stands for a more
general concept, where the ``sheaf of functions'' is an abstract
``sheaf of rings,'' not necessarily a subsheaf in the sheaf of all
functions on $M$. The above definition is simpler, although not as
standard.
\er

\exercise
Let $M,N$ be open subsets in $\Bbb R^n$ and let
$\Psi:M\to N$ be a smooth map. Show that $\Psi$ defines a morphism of
spaces ringed by smooth functions.
\ez

\exercise
Let $M$ be a smooth manifold of some class and let
${\cal F}$ be the space of functions of this class. Show that ${\cal F}$ is
a sheaf.
\ez

\exercise[!]
Let $M$ be a topological manifold, and let
$(U_i,\varphi_i)$ and $(V_j,\psi_j)$ be smooth structures on $M$. Show
that these structures are equivalent if and only if the corresponding
sheaves of smooth functions coincide.
\ez

\remark
This exercise implies that the following definition
is equivalent to the one stated earlier.
\er

\definition
Let $(M,{\cal F})$ be a topological manifold
equipped with a sheaf of functions. It is said to be a {\bf smooth
manifold} of {\bf class} $C^\infty$ or $C^i$ if every point in
$(M,{\cal F})$ has an open neighborhood isomorphic to the ringed space
$(\Bbb R^n,{\cal F}')$, where ${\cal F}'$ is a ring of 
functions on $\Bbb R^n$ of this class.
\ed

\definition
A {\bf coordinate system} on an open subset $U$
of a manifold $(M,{\cal F})$ is an isomorphism between $(U,{\cal F})$ and
an open subset in $(\Bbb R^n,{\cal F}')$, where ${\cal F}'$ are functions
of the same class on $\Bbb R^n$.
\ed

\remark
 In order to avoid complicated notation, from now on
we assume that all manifolds are Hausdorff and smooth (of class $C^\infty$). 
The case of other differentiability classes
can be considered in the same manner.
\er

\exercise[!]
Let $(M,{\cal F})$ and $(N,{\cal F}')$ be manifolds
and let $\Psi:M\to N$ be a continuous map. Show that the following
conditions are equivalent.

(i) In local coordinates, $\Psi$ is given by a smooth map

(ii) $\Psi$ is a morphism of ringed spaces.
\ez

\remark
An isomorphism of smooth manifolds is called a {\bf
diffeomorphism}. As follows from this
exercise, a diffeomorphism is a homeomorphism that maps smooth
functions onto smooth ones.
\er

\exercise[*]
Let ${\cal F}$ be a presheaf of functions on
$\Bbb R^n$. Figure out a minimal sheaf that contains ${\cal F}$ in the
following cases.
\enum
\item Constant functions.
\item Functions vanishing outside a bounded subset.
\item Bounded functions.
\ee
\ez

\exercise[*]
Describe all morphisms of ringed spaces from
$(\R^n,C^{i+1})$ to $(\R^n,C^i)$.
\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Embedded manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
 A {\bf closed embedding} $\phi:\; N\hookrightarrow M$ of
topological spaces is an injective map from $N$
to a closed subset $\phi(N)$ inducing a homeomorphism
of $N$ and $\phi(N)$. {\bf An open embedding}
$\phi:\; N\hookrightarrow M$ is a homeomorphism
of $N$ and an open subset of $M$. 
is an image of a closed embedding.
\ed

\definition
Let $M$ be a smooth manifold.
$N\subset M$ is called 
{\bf smoothly embedded submanifold
 of dimension $m$} if for every point $x\in N$, there is a
neighborhood $U\subset M$ diffeomorphic to an open ball $B\subset \R^n$, such that this
diffeomorphism maps $U\cap N$ onto a linear subspace of $B$
dimension $m$.
\ed




\exercise
Let $(M,{\cal F})$ be a smooth manifold and let
$N\subset M$ be a smoothly embedded submanifold. Consider the space
${\cal F}'(U)$ of smooth functions on $U\subset N$ that are extendable to
functions on $M$ defined on some neighborhood of $U$.
\enum
\ite
Show that ${\cal F}'$ is a sheaf.
\ite Show that this sheaf defines a smooth structure on $N$.
\ite Show that the natural embedding $(N,{\cal F}')\to(M,{\cal F})$ is a
morphism of manifolds.
\ee
\ez

\hint To prove that ${\cal F}$ is a sheaf, you might
need partition of unity introduced below. Sorry.
\eh

\exercise
 Let $N_1,N_2$ be two manifolds and let
$\varphi_i:N_i\to M$ be smooth embeddings. Suppose that the image of
$N_1$ coincides with that of $N_2$. Show that $N_1$ and $N_2$ are
isomorphic.
\ez

\remark
By the above problem, in order to define a smooth
structure on $N$, it suffices to embed $N$ into $\Bbb R^n$. As it will
be clear in the next handout, every manifold is embeddable into
$\Bbb R^n$ (assuming it admits partition of unity). Therefore, in place of a
smooth manifold, we can use ``manifolds that are smoothly embedded into
$\Bbb R^n$.''
\er

\exercise
 Construct a smooth embedding of
$\Bbb R^2/\Bbb Z^2$ into $\Bbb R^3$.
\ez

\exercise[**]
 Show that the projective space $\R P^n$ does not admit
a smooth embedding into $\Bbb R^{n+1}$ for $n>1$.
\ez


\NewVedomost


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Partition of unity}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
Show that an open ball ${\Bbb B}^n\subset \R^n$ is diffeomorphic to $\R^n$.
\ez

\definition
A cover $\{U_\alpha\}$ of a topological space $M$
is called {\bf locally finite} if every point in $M$ possesses a
neighborhood that intersects only a finite number of $U_\alpha$.
\ed

\exercise
Let $\{U_\alpha\}$ be a locally finite atlas on $M$, and
$U_\alpha\stackrel {\phi_\alpha}\arrow \R^n$ homeomorphisms.
Consider a cover $\{V_i\}$ of $\R^n$ given by open balls 
of radius $n$ centered in integer points, and let $\{W_\beta\}$
be a cover of $M$ obtained as union of $\phi_\alpha^{-1}(V_i)$.
Show that $\{W_\beta\}$ is locally finite.
\ez

\exercise
\label{_subcover_compact_Exercise_}
Let $\{U_\alpha\}$ be an atlas on a manifold $M$. 
\enum
\ite
Construct
a refinement $\{V_\beta\}$ of $\{U_\alpha\}$ such that
a closure of each $V_\beta$ is compact in $M$.
\ite
Prove that such a refinement can be chosen
locally finite if $\{U_\alpha\}$
is locally finite
\ee
\ez

\hint
Use the previous exercise.
\eh

\exercise
Let $K_1, K_2$ be non-intersecting compact subsets
of a Hausdorff topological space $M$. 
Show that
there exist a pair of open subsets $U_1\supset K_1$,
$U_2\supset K_2$ satisfying $U_1 \cap U_2=\emptyset$.
\ez

\exercise[!]
Let $U\subset M$ be an open subset with compact closure,
and $V\supset M\backslash U$ another open subset. Prove that
there exists $U'\subset U$ such that the closure of
$U'$ is contained in $U$, and $V\cup U'=M$.
\ez
\hint
Use the previous exercise
\eh

\definition
Let $U\subset V$ be two open subsets of $M$ such that the closure
of $U$ is contained in $V$. In this case we write
$U\Subset V$.
\ed

\exercise[!]
\label{_subcover_Subset_Exercise_}
Let $\{U_\alpha\}$ be a countable locally finite
cover of a Hausdorff topological space,
such that a closure of each $U_\alpha$ is compact.
Prove that there exists another cover
$\{V_\alpha\}$ indexed by the same set,
such that $V_\alpha\Subset U_\alpha$
\ez

\hint Use induction and the previous exercise.
\eh

\exercise[*]
Solve the previous exercise when $\{U_\alpha\}$ 
is not necessarily countable.
\ez

\hint
Some form of transfinite induction is required.
\eh


\exercise[!]
\label{_cover_subcover_Exercise_}
Denote by ${\Bbb B}\subset \R^n$ an open ball of radius 1.
Let $\{U_i\}$ be a locally finite 
countable atlas on a manifold $M$.
Prove that there exists a refinement 
$\{V_i, \phi_i:\; V_i \tilde\arrow \R^n\}$ 
of $\{U_i\}$ which is also locally finite,
and such that  $\bigcup_i\phi_i^{-1}({\Bbb B})=M$.
\ez

\hint 
Use Exercise \ref{_subcover_Subset_Exercise_} and
Exercise \ref{_subcover_compact_Exercise_}.
\eh

\definition
{\bf A function with compact support} is a function
which vanishes outside of a compact set.
\ed


\definition
 Let $M$ be a smooth manifold and let $\{U_\alpha\}$
be a locally finite cover of $M$. A {\bf partition of unity}
subordinate to the cover $\{U_\alpha\}$ is a family of smooth functions
$f_i:M\to[0,1]$ with compact support indexed by the same indices as the
$U_i$'s and satisfying the following conditions.

(a) Every function $f_i$ vanishes outside $U_i$

(b) $\sum_if_i=1$
\ed

\remark Note that the sum $\sum_if_i=1$
makes sense only when $\{U_\alpha\}$ is locally finite.
\er



\exercise
Show that all derivatives of $e^{-\frac1{x^2}}$ at
$0$ vanish.
\ez

\exercise
Define the following function $\lambda$ on
$\Bbb R^n$
$$\lambda(x):=\left\{\begin{matrix} e^{\frac1{|x|^2-1}}&\text{ if
}|x|<1\\0\hfill&\text{ if }|x|\ge1\end{matrix}\right.$$
Show that $\lambda$ is smooth and that all its derivatives vanish at
the points of the unit sphere.
\ez



\exercise
\label{_smooth_lambdas_Exercise_}
Let $\{U_i,\ \varphi_i:U_i\tilde\arrow\Bbb R^n\}$ be an
atlas on a smooth manifold $M$. Consider the following function
$\lambda_i:M\to[0,1]$
$$\lambda_i(m):=\left\{\begin{matrix}\lambda\big(\varphi_i(m)\big)&\text{ if
}m\in U_i\\0\hfill&\text{ if }m\notin U_i\end{matrix}\right.$$
Show that $\lambda_i$ is smooth.
\ez

\exercise[!]
{\bf (existence of partitions of unity)}\\
Let $\{U_i,\ \varphi_i:U_i\to\Bbb R^n\}$ be a
locally finite atlas on a manifold $M$ such that $\varphi_i^{-1}(B_1)$
cover $M$ as well (such an atlas was constructed in
Exercise \ref{_cover_subcover_Exercise_}).
Consider the functions $\lambda_i$'s constructed in 
Exercise \ref{_smooth_lambdas_Exercise_}. 
Show that $\sum_j\lambda_j$ is well defined,
vanishes nowhere, and that the family of
functions $\Big\{f_i:=\frac{\lambda_i}{\sum_j\lambda_j}\Big\}$ provides
a partition of unity on $M$.
\ez

\remark
From this exercise it follows that
any manifold with locally finite countable
atlas admits a partition of unity.
\er

\exercise[*]
Let $M$ be a manifold admitting
a countable atlas. Prove that $M$ admits
a countable, locally finite atlas, or find 
a counterexample.
\ez

\exercise[**]
Show that any Hausdorff, connected manifold admits
a countable, locally finite atlas, or find
a counterexample.
\ez

\exercise
Let $M$ be a compact manifold, 
$\{V_i, \phi_i:\; V_i \arrow \R^n, i=1, 2, ..., m\}$ 
an atlas (which can be chosen finite),
and $\nu_i:\; M \arrow [0,1]$ the subordinate partition of unity.
\enum
\ite[!] Consider the map $\Phi_i:\; M \arrow \R^{n+1}$, 
with \[ \Phi_i(z):= 
\frac{(\nu_i\phi_i(z), 1)}
{|\nu_i\phi_i(z)|^2 +1} 
\]
Show that $\Phi_i$ is smooth, and its image lies in
the $n$-dimensional sphere $S^{n}\subset \R^{n+1}$.
\ite[*] Show that $\Phi_i:\; M \arrow S^{n}$ is
surjective.
\ite[!] Let $U_i\subset V_i$ be the set where $\nu_i\neq 0$.
Show that the restriction $\Phi_i\restrict{V_i}:\; V_1 \arrow S^n$
is an open embedding.
\ite[!] Show that $\prod_{i=1}^m:\;\Phi_i:\; M \arrow \underbrace{S^n\times
S^n \times ...\times S^n}_{\text{$m$ times}}$
is a closed embedding.
\ee
\ez

\remark We have just proved a weaker form of Whitney's theorem:
each compact manifold admits a smooth embedding to $\R^N$.
\er











\end{document}
