% figure out how to use san serif in AMS-TeX
\input amstex
\documentstyle{amsppt}\nologo\footline={}\subjclassyear{2000}

\def\diam{\mathop{\text{\rm diam}}}
\def\Id{\mathop{\text{\rm Id}}}
\def\Vol{\mathop{\text{\rm Vol}}}
\def\T{\mathop{\text{\rm T}}}

\hsize450pt\vsize590pt\topmatter\title Analysis 2: Hausdorff
dimension and Whitney's theorem.\endtitle\author Misha
Verbitsky\endauthor\address\endaddress\email\endemail
%\subjclass\endsubjclass\abstract\endabstract
\endtopmatter\document

In order to be approved, you should solve in every sheet either
all problems with asterisks or all problems without asterisks. The
problems with two asterisks are optional: $k$ problems with two
asterisks substitute $2k$ problems with one asterisk. The problems
with (!) are obligatory for everybody.

\bigskip

\noindent
{\bf2.1.~Hausdorff dimension}

\medskip

\noindent
{\bf Definition 2.1.} Let $M$ be a metric space. The {\bf diameter}
$\diam M\in[0,\infty]$ is the number $\sup\limits_{x,y\in M}d(x,y)$.

\smallskip

\noindent {\bf Definition 2.2.} In a metric space, the {\bf ball} of
radius $\varepsilon$ centered at $x$ is defined as the set of all
points $y$ satisfying $d(x,y)<\varepsilon$.

\smallskip

\noindent
{\bf Problem 2.1.} Describe all possible values of the diameter of the
ball of radius $\varepsilon$ in a metric space.

\smallskip

\noindent
{\bf Problem 2.2.} Let $M$ be a metric space and let $\varepsilon>0$.
Show that $M$ possesses a cover by balls of diameter $\le\varepsilon$.

\smallskip

\noindent
{\bf Definition 2.3.} Let $\{S_i\}$ be a cover of a metric space $M$
formed by balls of radius $r$ with $r<\varepsilon$. Define
$\mu_{d,\varepsilon}\in[0,\infty]$ as
$$\mu_{d,\varepsilon}M:=\inf_{\{S_i\}}\sum_i(\diam S_i)^d,$$
where the infimum is taken with respect to all covers as above. The
limit
$$\mu_dM:=\sup\lim_{\varepsilon\to0}\mu_{d,\varepsilon}M$$
is called $d$-{\bf dimensional Hausdorff measure} of $M$.

\smallskip

\noindent
{\bf Problem 2.3.} Suppose that a metric in $M=\Bbb R^n$ is given by
the norm $\big|(x_1,\dots,x_n)\big|:=\max|x_i|$. Show that the
$n$-dimensional Hausdorff measure of a polyhedron equals its volume (in
the usual sense).

\smallskip

\noindent
{\bf Problem 2.4 (*).} Suppose that a metric in $M=\Bbb R^n$ is given
by the norm $\big|(x_1,\dots,x_n)\big|:=\sum|x_i|$. Show that the
$n$-dimensional Hausdorff measure of a polyhedron is proportional to
its volume. Calculate the coefficient of proportionality.

\smallskip

\noindent
{\bf Problem 2.5 (*).} Let $M=\Bbb R^n$ be equipped with the euclidean
metric. Show that the $n$-dimensional Hausdorff measure of a polyhedron
is proportional to its volume. Calculate the coefficient of
proportionality.

\smallskip

\noindent
{\bf Definition 2.4.} A map $f:M\to N$ of metric spaces is called
lipschitz with constant $C\ge0$ if
$d(x,y)\ge C\cdot d\big(f(x),f(y)\big)$ for all $x,y\in M$. A map is
called bilipschitz if it is bijective and the inverse map is also
lipschitz (with some constant).

\smallskip

\noindent
{\bf Problem 2.6.} Show that every lipschitz map is continuous.

\smallskip

\noindent
{\bf Problem 2.7 (*).} Construct an example of a continuous map of
metric spaces that is not lipschitz.

\smallskip

\noindent
{\bf Problem 2.8.} Let $d_1,d_2$ be two norms on a vector space $V$.
Denote by the same letters the corresponding metrics. Show that the
identity map $\Id_V:(V,d_1)\to(V,d_2)$ is lipschitz if and only if the
unit ball $B_1(r,d_1)$ is limited in terms of the norm $d_2$.

\smallskip

\noindent
{\bf Problem 2.9 (*).} Let $M=\Bbb R^n$ and let $d_1,d_2$ be some
norms on $M$. Show that $\Id_M:(M,d_1)\to(M,d_2)$ is bilipschitz.

\smallskip

\noindent
{\bf Problem 2.10 (!).} Let $U\subset\Bbb R^n$ be a limited open subset
and let $\Phi:U\to\Bbb R^n$ be a smooth map smoothly extendable to the
boundary $\partial U$. Show that $\Phi$ is lipschitz.

\smallskip

\noindent
{\bf Problem 2.11.} Let $M\overset f\to\longrightarrow N$ be a
lipschitz map of metric spaces with constant $C$. Show that
$\mu_dM\ge C^d\mu_df(M)$, where $\mu_d$ stands for the $d$-dimensional
Hausdorff measure.

\smallskip

\noindent
{\bf Problem 2.12 (!).} Suppose that $\mu_dM<\infty$. Show that
$\mu_{d'}M=0$ for every $d'>d$.

\smallskip

\noindent
{\bf Hint.} Deduce from $\diam S_i<\varepsilon$ the inequality
$$\mu_{d',\varepsilon}M=\inf\limits_{\{S_i\}}\sum\limits_i
(\diam S_i)^{d'}\le\varepsilon^{d'-d}\inf\limits_{\{S_i\}}\sum\limits_i
(\diam S_i)^d=\varepsilon^{d'-d}\mu_{d,\varepsilon}M\eqno{(1)}$$
and pass to the limit $\varepsilon\to0$.

\smallskip

\noindent
{\bf Problem 2.13 (!).} Suppose that $\mu_{d'}M=\infty$. Show that
$\mu_dM=0$ for every $d<d'$.

\smallskip

\noindent
{\bf Hint.} Use the inequality (1) and pass to the limit
$\varepsilon\to0$.

\smallskip

\noindent
{\bf Definition 2.5.} Let $M$ be a metric space. The {\bf Hausdorff
dimension} $\dim_HM\in[0,\infty]$ is the supremum of all $d$ such that
$\mu_dM=\infty$.

\smallskip

\noindent
{\bf Problem 2.14.} Find the Hausdorff dimension of a finite set.

\smallskip

\noindent
{\bf Problem 2.15.} Let $f:M\to N$ be a lipschitz map. Show that $f$
does not increase the Hausdorff dimension: $\dim_HM\ge\dim_Hf(M)$.

\smallskip

\noindent
{\bf Problem 2.16.} Show that every bilipschitz map preserve Hausdorff
dimension (``Hausdorff dimension is a bilipschitz invariant'').

\smallskip

\noindent
{\bf Problem 2.17 (*).} Find the Hausdorff dimension of the Cantor set
$K\subset[0,1]$.

\smallskip

\noindent
{\bf Definition 2.6.} A subset $Z\subset\Bbb R^n$ has {\bf measure
zero} if, for every $\varepsilon>0$, there exists a countable cover of
$Z$ by balls $U_i$ such that $\sum_i\Vol U_i<\varepsilon$.

\smallskip

\noindent
{\bf Problem 2.18.} Show that the countable union of subsets of zero
measure has measure zero.

\smallskip

\noindent
{\bf Problem 2.19.} Show that the image of a subset of zero measure
under a lipschitz map $\Bbb R^n\to\Bbb R^n$ has measure zero.

\smallskip

\noindent
{\bf Problem 2.20 (!).} Show that the image of a subset of zero measure
under a smooth map $\Bbb R^n\to\Bbb R^n$ has measure zero.

\smallskip

\noindent
{\bf Problem 2.21 (!).} Construct an example of a continuous map from
$\Bbb R^n$ to $\Bbb R^n$ that sends a subset of zero measure onto a
subset of nonzero measure.

\smallskip

\noindent
{\bf Problem 2.22 (!).} Let $M\subset\Bbb R^d$ be a subset such that
$\dim_HM<d$. Show that $M$ has measure zero.

\smallskip

\noindent
{\bf Definition 2.7.} Let $M$ be a smooth manifold with an atlas
$\{U_i,\ \varphi_i:U_i\to\Bbb R^n\}$. A subset $Z\subset M$ has {\bf
measure zero} if the image $\varphi(Z\cap U_i)$ has measure zero in
$\Bbb R^n$ for every $i$.

\smallskip

\noindent
{\bf Problem 2.23.} Show that this definition does not depend on the
choice of an atlas on $M$.

\smallskip

\noindent
{\bf Problem 2.24.} Let $M\overset f\to\longrightarrow\Bbb R^n$ be a
smooth map of manifolds and let $M$ be a union of compact subsets. Show
that $\dim_Hf(M)\le\dim M$.

\smallskip

\noindent
{\bf Hint.} Show first that $f$ is lipschitz on compact subsets. Then
use the fact that lipschitz maps satisfy $\dim_Hf(M)\le\dim M$.

\smallskip

\noindent
{\bf Problem 2.25 (!).} Let $M\overset f\to\longrightarrow N$ be a
smooth map of manifolds such that $\dim M<\dim N$. Show that the image
of $M$ has measure zero.

\smallskip

\noindent
{\bf Hint.} Use the previous problem.

\smallskip

\noindent
{\bf Remark.} This theorem is a particular case of Sard's theorem that
claims that the set of critical values of a smooth map has measure
zero.

\smallskip

\noindent
{\bf Problem 2.26 (**).} Deduce Sard's theorem from the above problem.

\bigskip

\noindent
{\bf2.2.~Whitney's theorem (with a bound on dimension)}

\medskip

\noindent
{\bf Definition 2.8.} A smooth map of manifolds
$M\overset f\to\longrightarrow N$ is called {\bf immersion} if the
differential $\Cal Df$ is an embedding in local coordinates.

\smallskip

\noindent
{\bf Definition 2.9.} The {\bf Klein bottle} is the quotient of
the two-dimensional torus $T^2:=\Bbb S^1\times\Bbb S^1$ by the action
of the group $\Bbb Z/2\Bbb Z$ mapping $(t_1,t_2)$ to $(t_1+\pi,-t_2)$.

\smallskip

\noindent
{\bf Problem 2.27.} Show that the indicated action is free and that the
quotient is a manifold.

\smallskip

\noindent
{\bf Problem 2.28.} Construct an immersion of the Klein bottle into
$\Bbb R^3$.

\smallskip

\noindent
{\bf Problem 2.29 (!).} Let $M\overset f\to\longrightarrow N$ be a
smooth map of manifolds. Show that $f$ is a smooth embedding if and
only if it is an injective immersion.

\smallskip

\noindent
{\bf Hint.} Use the inverse function theorem.

\smallskip

\noindent
{\bf Definition 2.10.} Let $M\hookrightarrow\Bbb R^n$ be a smooth
$m$-submanifold. The {\bf tangent plane} at $p\in M$ is the plane in
$\Bbb R^n$ tangent to $M$ (i.e, the plane lying in the image of the
differential given in local coordinates). A {\bf tangent vector} is an
arbitrary vector in this plane with the origin at $p$. The space of all
tangent vectors at $p$ is denoted by $\T_pM$. When a metric on
$\Bbb R^n$ is given, we can define the space of {\bf unit tangent
vectors} $\Bbb S^{m-1}M$ as the set of all pairs $(p,v)$, where
$p\in M$, $v\in\T_pM$, and $|v|=1$.

\smallskip

\noindent
{\bf Problem 2.30.} Show that $\Bbb S^{m-1}M$ is a manifold and that
the natural projection $\Bbb S^{m-1}M\to M$ is a smooth map with fibers
$\Bbb S^{m-1}$.

\smallskip

\noindent
{\bf Remark.} $\Bbb S^{m-1}M$ is called the {\bf unit sphere bundle}
over $M$.

\smallskip

\noindent
{\bf Problem 2.31 (*).} Show that $\Bbb S^{m-1}M$ is independent of
an embedding $M\hookrightarrow\Bbb R^n$, i.e., for two different
embeddings of $M$ into $\Bbb R^n$ and into $\Bbb R^{n'}$, the
corresponding manifolds $\Bbb S^{m-1}M$ are diffeomorphic.

\smallskip

\noindent
{\bf Problem 2.32 (!).} Let
$M\overset\varphi\to\hookrightarrow\Bbb  R^n$ be a manifold of
dimension $m$ embedded into $\Bbb R^n$, let
$\lambda\in\Bbb P_\Bbb R^{n-1}$ be a straight line in $\Bbb R^n$, and
let $P_\lambda:\Bbb R^n\to\Bbb R^{n-1}$ denote the projection onto the
quotient $\Bbb R^n/\lambda\cong\Bbb R^{n-1}$.

a. Let $\Delta\subset M\times M$ stand for the diagonal. Define the map
$M\times M\setminus\Delta\overset B\to\longrightarrow\Bbb P_\Bbb
R^{n-1}$
by sending the pair of points $(x,y)\in M\times M$ to the straight line
passing through $\varphi(x)-\varphi(y)$. Show that
$P_\lambda\circ\varphi:M\to\Bbb R^{n-1}$ is an injection if and only
if $\lambda$ does not lie in the image of $B$.

b. Define the map
$\Bbb S^{m-1}M\overset B_0\to\longrightarrow\Bbb P_\Bbb R^{n-1}$ by
sending a tangent vector to the corresponding straight line. Show that
$P_\lambda\circ\varphi:M\to\Bbb R^{n-1}$ is an immersion if and only if
$\lambda$ does not lie in the image of $B_0$.

\smallskip

\noindent
{\bf Problem 2.33 (!).} Let
$M\overset\varphi\to\hookrightarrow\Bbb  R^n$ be an embedded manifold
of dimension $m$ with $n>2m+2$. Show that there exists a projection
$\Bbb R^n\overset P\to\longrightarrow\Bbb R^{2m+2}$ such that
$P\circ\varphi:M\to\Bbb R^{2m+2}$ is an immersion.

\smallskip

\noindent
{\bf Hint.} Use the fact that the images of the maps $B_0$ and $B$ in
the previous problem have measure zero and apply induction on $n$.

\smallskip

\noindent
{\bf Problem 2.34.} Under the conditions of the previous problem, show
that there exists a projection
$\Bbb R^n\overset P\to\longrightarrow\Bbb R^{2m+1}$ such that
$P\circ\varphi:M\to\Bbb R^{2m+1}$ is an immersion.

\smallskip

\noindent
{\bf Problem 2.35.} Is any $n$-dimensional manifold embeddable in
$\Bbb R^{2n-1}$ ?

\smallskip

\noindent
{\bf Problem 2.36 (**).} Is it possible to construct an immersion of
the projective space $\Bbb P_\Bbb C^2$ into $\Bbb R^5$ ?

\smallskip

\noindent
{\bf Problem 2.37.} Let $M$ be a compact Hausdorff manifold of
dimension $n$. Show that $M$ admits a smooth closed embedding into
$\Bbb R^{2n+2}$.

\smallskip

\noindent
{\bf Remark.} Whitney showed that any Hausdorff $m$-dimensional
manifold with a countable basis of topology admits a closed embedding
into $\Bbb R^{2m}$. This statement is called the ``strong Whitney
theorem.''

\bigskip

\noindent
{\bf2.3.~Whitney's theorem (for noncompact manifolds)}

\medskip

\noindent
{\bf Problem 2.38.} Let $\Cal P$ denote the space of embeddings
$\Bbb R^m\to\Bbb R^{2m+2}$ equipped with the natural topology. Show
that $\Cal P$ is a manifold. Construct a smooth structure on $\Cal P$.

\smallskip

\noindent
{\bf Problem 2.39.} Let $M$ be a manifold with a countable basis.

a. Show that $M$ is a union of an ascending countable chain of
compact subsets.

b. Show that $M$ admits a partition of unity.

\smallskip

\noindent
{\bf Problem 2.40.} Let $M$ be an $n$-dimensional manifold, let
$\{U_i,\ \varphi_i:U_i\to\Bbb R^n\}$ be a locally finite atlas, and let
$f_i:U_i\to[0,1]$ be a corresponding partition of unity. Consider the
map $\Psi_i:M\to\Bbb R^{n+1}$ constructed as in the previous sheet
$$\Psi_i(m):=\left\{\matrix\big(f_i(m)\varphi_i(m),f_i(m)\big)&\text{
if }m\in U_i\,\,\\(0,\dots,0)\hfill&\text{ if }m\notin
U_i.\endmatrix\right.$$
Let $A_i\in\Cal P$ be a family of embeddings $\Bbb R^n\to\Bbb R^{2n+2}$
with the same set of indices. Consider the map
$\Psi_A:M\to\Bbb R^{2n+2}$, $\Psi_A(m):=\sum A_i\big(\Psi_i(m)\big)$.
Show that this map is well defined. Show that it can be obtained as a
composition of the embedding $\bigoplus_i\Psi_i:M\to\Bbb R^\infty$ and
a linear projection $\Bbb R^\infty\to\Bbb R^{2n+2}$.

\smallskip

\noindent
{\bf Problem 2.41 (*).} Under the conditions of the previous problem,
let $M_0\subset M$ be a compact subset and let
$\bigcup_{i\in I}U_i\supset M_0$ be a corresponding finite subcover in
$\{U_i\}$ of $k$ members. Show that there exists a subset
$Z_I\subset\Cal P^k$ of zero measure such that, for all collections
$\{A_i,\ i\in I\}\in\Cal P^k$ not belonging to $Z_I$, the corresponding
map $\Psi_A:M_0\to\Bbb R^{2n+2}$ is a smooth embedding.

\smallskip

\noindent
{\bf Hint.} Use the proof of the Whitney theorem for compact $M$ given
in the previous section.

\smallskip

\noindent
{\bf Problem 2.42 (*).} Denote by $\Cal P^\infty$ the product of
$\Cal P$ with respect to the same set of indices as the one used in the
atlas $\{U_i\}$. Consider $\Cal P^\infty$ equipped with the product
Lebesgue measure. Show that the set $Z$ of all $\{A_i\}\in\Cal P$ such
that $\Psi_A$ is not an embedding has measure zero in $\Cal P^\infty$.

\smallskip

\noindent
{\bf Hint.} By construction, $Z$ is the union of all inverse images of
the sets $Z_I\subset\Cal P^k$ constructed in Problem~2.41 under the
standard projection
$\Cal P^\infty\overset\Pi_I\to\longrightarrow\Cal P^k$. Every such
inverse image has measure zero, hence, $Z$, being a union of subsets
of zero measure, has measure zero.

\enddocument