\documentclass{slides} \usepackage{amssymb, amsmath, amscd, color, epsfig} %\usepackage[matrix,arrow]{xy} \newcommand{\green}{\color[rgb]{0,0.4,0}} \newcommand{\purple}{\color[rgb]{0.4,0,0.4}} \newcommand{\red}{\color[rgb]{0.7,0,0}} \newcommand{\blue}{\color{blue}} \def\eqref#1{(\ref{#1})} \newcommand{\goth}{\mathfrak} \newcommand{\g}{{\frak g}} \newcommand{\arrow}{{\:\longrightarrow\:}} \newcommand{\Z}{{\Bbb Z}} \newcommand{\C}{{\Bbb C}} \newcommand{\R}{{\Bbb R}} \newcommand{\Q}{{\Bbb Q}} \renewcommand{\H}{{\Bbb H}} \newcommand{\6}{\partial} \def\1{\sqrt{-1}\:} \newcommand{\restrict}[1]{{\left|_{{#1}}\right.}} \newcommand{\cntrct} % contraction with a vector field {\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}} \def\Bbb#1{\mathbb #1} \newcommand{\calo}{{\cal O}} \newcommand{\cac}{{\cal C}} % Correcting TeX... %\let\oldtilde=\tilde %\renewcommand{\tilde}{\widetilde} \renewcommand{\bar}{\overline} \renewcommand{\phi}{\varphi} \renewcommand{\epsilon}{\varepsilon} \renewcommand{\geq}{\geqslant} \renewcommand{\leq}{\leqslant} % Operatornames \newcommand{\even}{{\rm even}} \newcommand{\ev}{{\rm even}} \newcommand{\odd}{{\rm odd}} \newcommand{\const}{{\it const}} \newcommand{\fl}{{\rm fl}} \newcommand{\im}{\operatorname{im}} \newcommand{\End}{\operatorname{End}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\Hol}{\operatorname{{\cal H}ol}} \newcommand{\Tot}{\operatorname{Tot}} \newcommand{\Id}{\operatorname{Id}} \newcommand{\id}{\operatorname{\text{\sf id}}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Alt}{\operatorname{Alt}} \newcommand{\Iso}{\operatorname{Iso}} \newcommand{\Sec}{\operatorname{Sec}} \newcommand{\Can}{\operatorname{Can}} \newcommand{\Sing}{\operatorname{Sing}} \newcommand{\Spin}{\operatorname{Spin}} \newcommand{\codim}{\operatorname{codim}} \newcommand{\coim}{\operatorname{coim}} \newcommand{\coker}{\operatorname{coker}} \newcommand{\slope}{\operatorname{slope}} \newcommand{\rk}{\operatorname{rk}} \newcommand{\Def}{\operatorname{Def}} \newcommand{\Lie}{\operatorname{Lie}} \newcommand{\Tw}{\operatorname{Tw}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\Diff}{\operatorname{Diff}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\inbfpare}[1]{{% \mbox{\tt (}\hspace{-5pt}\mbox{\tt (} #1 % \mbox{\tt )}\hspace{-5pt}\mbox{\tt )}% }} \newcommand{\comment}[1]{{}} \def\blacksquare{\hbox{\vrule width 10pt height 10pt depth 0pt}} \def\endproof{\blacksquare} \def\shortdash{\mbox{\vrule width 4.5pt height 0.55ex depth -0.5ex}} \makeatletter %\@ifundefined{Bbb} % {\newcommand{\Bbb}[1]{{\mathbb #1}}}% %{}% {\edef\Bbb#1{{\Bbb #1}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Pagestyle % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\ps@verbit}{% \renewcommand{\@oddhead}{% \scriptsize {\it \small Geometry of manifolds, lecture 2 \hfil \tiny M. Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \setlength\paperheight {10in}% \setlength\paperwidth {13.5in} \setlength{\textwidth}{0.8\paperwidth} \setlength{\textheight}{0.8\paperheight} \setlength{\pdfpageheight}{\paperheight} \setlength{\pdfpagewidth}{\paperwidth} \addtolength{\topmargin}{-20mm} \addtolength{\leftmargin}{-25mm} \addtolength{\rightmargin}{-25mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Lemma, sublemma, corollary, proposition, theorem, % % definition,example defined there: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcounter{section} \newcounter{Mycounter}[section] \newcounter{lemma}[section] \setcounter{lemma}{0} \renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}} \newcommand{\lemma}{% \setcounter{lemma}{\value{Mycounter}} \refstepcounter{lemma} \stepcounter{Mycounter} {\bf \green LEMMA:\ }} \newcounter{claim}[section] \setcounter{claim}{0} \renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}} \newcommand{\claim}{% \setcounter{claim}{\value{Mycounter}} \refstepcounter{claim} \stepcounter{Mycounter} {\bf \green CLAIM:\ }} \newcounter{corollary}[section] \setcounter{corollary}{0} \renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}} \newcommand{\corollary}{% \setcounter{corollary}{\value{Mycounter}} \refstepcounter{corollary} \stepcounter{Mycounter} {\bf \green COROLLARY:\ }} \newcounter{theorem}[section] \setcounter{theorem}{0} \renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}} \newcommand{\theorem}{% \setcounter{theorem}{\value{Mycounter}} \refstepcounter{theorem} \stepcounter{Mycounter} {\bf \green THEOREM:\ }} \newcounter{conjecture}[section] \setcounter{conjecture}{0} \renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}} \newcommand{\conjecture}{% \setcounter{conjecture}{\value{Mycounter}} \refstepcounter{conjecture} \stepcounter{Mycounter} {\bf \green CONJECTURE:\ }} \newcounter{proposition}[section] \setcounter{proposition}{0} \renewcommand{\theproposition} {\noindent{Proposition \thesection.\arabic{proposition}}} \newcommand{\proposition}{% \setcounter{proposition}{\value{Mycounter}} \refstepcounter{proposition} \stepcounter{Mycounter} {\bf \green PROPOSITION:\ }} \newcounter{definition}[section] \setcounter{definition}{0} \renewcommand{\thedefinition} {\noindent{Definition~\thesection.\arabic{definition}}} \newcommand{\definition}{% \setcounter{definition}{\value{Mycounter}} \refstepcounter{definition} \stepcounter{Mycounter} {\bf \green DEFINITION:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \green EXAMPLE:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \green REMARK:\ }} \newcounter{observation}[section] \setcounter{observation}{0} \renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcounter{question}[section] \setcounter{question}{0} \renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}} \newcommand{\question}{% \setcounter{question}{\value{Mycounter}} \refstepcounter{question} \stepcounter{Mycounter} {\bf \green QUESTION:\ }} \newcommand{\exercise}{{\bf \green EXERCISE:\ }} \newcommand{\proof}{{\bf \green Proof:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Geometry of manifolds \\[15mm] \small lecture 2: manifolds and ringed spaces} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf Math in Moscow and HSE \\[2mm] February 11, 2013 } \end{center} \newpage {\bf \blue Sheaves and exact sequences} \definition {\bf\blue An open cover} of a topological space $X$ is a family of open sets $\{U_i\}$ such that $\bigcup_iU_i=X$. \remark {\bf \purple A presheaf of functions} is a collection of subrings of functions on open subsets, compatible with restrictions. {\bf\purple A sheaf of fuctions is a presheaf allowing ``gluing''} a function on a bigger open set if its restrictions to smaller open sets are compatible. \definition A sequence $A_1 \arrow A_2 \arrow A_3 \arrow ...$ of homomorphisms of abelian groups or vector spaces is called {\bf\blue exact} if the image of each map is the kernel of the next one. \claim A presheaf ${\cal F}$ is a sheaf if and only if for every cover $\{U_i\}$ of an open subset $U\subset M$, {\bf \red 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 \[ \eta\restrict{U_i\cap U_j}-\eta\restrict{U_j\cap U_i} \in {\cal F}(U_i\cap U_j)\oplus {\cal F}(U_j\cap U_i)\subset \prod\limits_{i\ne j}{\cal F}(U_i\cap U_j) \] \newpage {\bf \blue Ringed spaces} A {\bf\blue ringed space} $(M,{\cal F})$ is a topological space equipped with a sheaf of functions. A~{\bf \blue 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 $\psi^* f:=f\circ\Psi$ belongs to the ring ${\cal F}\big(\Psi^{-1}(U)\big)$. An {\bf\blue isomorphism} of ringed spaces is a homeomorphism $\Psi$ such that $\Psi$ and $\Psi^{-1}$ are morphisms of ringed spaces. \definition Let $(M,{\cal F})$ be a topological manifold equipped with a sheaf of functions. It is said to be a {\bf\blue smooth manifold of class} $C^\infty$ or $C^i$ if every point in $(M,{\cal F})$ has an open neighborhood isomorphic to the ringed space $({\Bbb B}^n,{\cal F}')$, where ${\Bbb B}^n\subset \R^n$ is an open ball and ${\cal F}'$ is a ring of functions on an open ball ${\Bbb B}^n$ of this class. \definition {\bf\blue Diffeomorphism} of smooth manifolds is a homeomorphism $\phi$ which induces an isomorphims of ringed spaces, that is, $\phi$ and $\phi^{-1}$ map (locally defined) smooth functions to smooth functions. {\bf \red Assume from now on that all manifolds are Hausdorff and of class $C^\infty$}. \newpage {\bf \blue Charts and coordinates} \definition {\bf \blue Coordinate system} on a manifold $M$ is an open subset $V\subset M$ equipped with an isomorphism of ringed spaces $(V, C^\infty V)\cong({\Bbb B}^n, C^\infty{\Bbb B}^n)$ per definition of a manifold. \definition {\bf \blue A chart} on a smooth manifold $(M, C^\infty M)$ is an open subset $U\subset M$ together with an embedding $\psi:\; U \arrow \R^n$ given by smooth functions $\phi_1, ..., \phi_n\in C^\infty M$ inducing a diffeomorphism on any open subset $V\subset U$ equipped with a coordinate system $(V, C^\infty V)\cong ({\Bbb B}^n, C^\infty{\Bbb B}^n)$. \definition {\bf \blue Transition map} between two charts $\psi_1:\; U_1 \arrow \R^n$ and $\psi_2:\; U_2 \arrow \R^n$ is a map $\Psi_{ij}:\;\psi_1(U_1\cap U_2) \arrow \psi_2(U_1\cap U_2)$ defined as $\Psi_{ij}(x)= \psi_2(\psi_1^{-1}(x))$. \claim {\bf \red Transition maps are smooth.} \proof In local coordinates all functions $\phi_1, ..., \phi_n$ used in the definition of the transition map are smooth. \endproof \newpage {\bf \blue Atlases on manifolds} \definition {\bf \blue An atlas} on a smooth manifold is a collection of charts $\{U_i, \psi_i:\; U_i \arrow \R^n\}$ satisfying $\bigcup U_i =M$ together with their transition maps. \remark In such a situation, the {\bf \purple charts $U_i$ are usually identified with their images $\psi(U_i)\subset \R^n$.} \remark {\bf \purple The sheaf $C^\infty M$ can be reconstructed from an atlas} as follows: a function $f$ on $U\subset M$ is smooth if and only if its restrictions to $U\cap U_i$ are smooth on all charts. \newpage {\bf \blue Embedded submanifolds} \definition A {\bf\blue 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 $\phi(N)$. \definition $M\subset \R^n$ is called {\bf\blue a submanifold} of dimension $m$ if for every point $x\in N$, there is a neighborhood $U\subset \R^n$ diffeomorphic to an open ball, such that this diffeomorphism maps $U\cap N$ onto a linear subspace of dimension $m$. \definition Let $M\subset \R^n$ be a submanifold. {\bf\blue A sheaf of smooth functions} $C^\infty M$ is defined as the sheaf of all functions on $U\subset M$ which can be locally extended to smooth functions on an open subset of $\R^n$ containing $U$. \newpage {\bf \blue Embedded submanifolds are smooth} \example Let $M=\R^k\subset \R^n$ be a linear space, and $\pi:\; \R^n\arrow M$ be a linear projection. {\bf \purple A function $f$ on $U\subset M$ is smooth if and only if $\pi^* f$ is smooth} (by definition, $\pi^*f(z)=f(\pi(z))$). Therefore, {\bf \red any smooth function on $U$ can be extended to $\pi^{-1}(U)\subset\R^n$.} We obtain that $(M,C^\infty M)\cong (\R^k, C^\infty \R^k)$. \claim Let $M\subset \R^n$ be a submanifold, and $C^\infty M$ the sheaf defined above. {\bf \red Then $(M, C^\infty M)$ is a smooth manifold.} \proof Locally around each point of $M$, the pair $(\R^n, M)$ is diffeomorphic to $(\R^n, \R^k)$. Then the previous example can be applied. \endproof \theorem {\bf \red Any manifold can be embedded to $\R^n$.} Its proof takes some work and will be done in the next lecture. \newpage {\bf \blue Locally finite covers} \definition An open cover $\{U_\alpha\}$ of a topological space $M$ is called {\bf\blue locally finite} if every point in $M$ possesses a neighborhood that intersects only a finite number of $U_\alpha$. \claim Let $\{U_\alpha\}$ be a locally finite atlas on a manifold $M$. {\bf \purple Then there exists a refinement $\{V_\beta\}$ of $\{U_\alpha\}$ such that a closure of each $V_\beta$ is compact in $M$.} {\bf \green Proof:} 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 2 centered in integer points, and let $\{W_\beta\}$ be a cover of $M$ obtained as union of $\phi_\alpha^{-1}(V_i)$. {\bf \purple Then $\{W_\beta\}$ is locally finite.} \endproof \definition Let $U\subset V$ be two open subsets of $M$ such that the closure of $U$ is contained in $V$. {\bf \blue In this case we write $U\Subset V$.} \newpage {\bf \blue Locally finite covers and their subcovers} \exercise Let $K_1, K_2$ be non-intersecting closed subsets of a Hausdorff topological space $M$, with $K_1$ compact. {\bf \red Show that there exist a pair of open subsets $U_1\supset K_1$, $U_2\supset K_2$ satisfying $U_1 \cap U_2=\emptyset$.} \claim Let $U\subset M$ be an open subset with compact closure, and $V\supset M\backslash U$ another open subset. {\bf \purple Then there exists $U'\subset U$ such that the closure of $U'$ is contained in $U$, and $V\cup U'=M$.} {\bf \green Proof. Step 1:} The complement $M\backslash U$ does not intersect $M\backslash V$, these sets are closed, and $M\backslash V$ is compact. {\bf \green Step 2:} Take open subsets $A, B \subset M$ separating $M\backslash U$ and $M\backslash V$. Then the closure $\bar B$ does not intersect $M\backslash U$, because $B$ does not intersect an open neighbourhood of $M\backslash U$. Therefore, $U':=B$ lies in $U$ with its closure. {\bf \green Step 3:} By construction, $\bar B$ contains $M\backslash V$, hence $U'\cup V=M$. \endproof \newpage {\bf \blue Locally finite covers and their subcovers} \theorem Let $\{U_\alpha\}$ be a countable locally finite cover of a Hausdorff topological space, such that a closure of each $U_\alpha$ is compact. {\bf \red Then there exists another cover $\{V_\alpha\}$ indexed by the same set, such that $V_\alpha\Subset U_\alpha$.} {\bf \green Proof. Step 1:} Let $U_1, U_2, ...$ be all elements of the cover. Suppose that $V_1, ..., V_{n-1}$ is already found. To take an induction step it remains to find $V_n \Subset U_n$ {\bf \green Step 2:} Replacing $U_i$ by $V_i$ and renumbering, we may assume that $n=1$. {\bf \purple Then the statement of Theorem follows from the previous exercise applied to $V=\bigcup_{i=2}^\infty U_i$ and $U=U_1$.} \endproof \newpage {\bf \blue Construction of a partition of unity} \remark If all $U_\alpha$ are diffeomorphic to $\R^n$, all $V_\alpha$ can be chosen diffeomorphic to an open ball. Indeed, any compact set is contained in an open ball. \corollary Let $M$ be a manifold admitting a locally finite countable cover $\{U_\alpha\}$, with $\phi_\alpha:\; U_\alpha \arrow \R^n$ diffeomorphisms. {\bf \purple Then there exists another atlas $\{U_\alpha, \phi_\alpha':\; U_\alpha \arrow \R^n\}$, such that $\phi'_\alpha({\Bbb B})$ is also a cover of $M$, and ${\Bbb B}\subset \R^n$ a unit ball.} \endproof \exercise {\bf \purple Find a smooth function $\nu:\; \R^n \arrow [0,1]$ which vanishes outside of ${\Bbb B}\subset \R^n$ and is positive on ${\Bbb B}$.} \remark In assumptions of Corollary, let $\nu_\alpha(z):= \nu(\phi_\alpha')$, and $\mu_i:=\frac{\nu_i}{\sum_\alpha\nu_\alpha}$. Then $\mu_\alpha:\; M \arrow [0,1]$ are smooth functions with support in $U_\alpha$ satisfying $\sum_\alpha \mu_\alpha=1$. Such a set of functions is called {\bf \blue a partition of unity}. \newpage {\bf \blue Partition of unity: a formal definition} \definition Let $M$ be a smooth manifold and let $\{U_\alpha\}$ a locally finite cover of $M$. A {\bf \blue 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) $\sum_if_i=1$ The argument of previous page proves the following theorem. \theorem Let $\{U_\alpha\}$ be a countable, locally finite cover of a manifold $M$, with all $U_\alpha$ diffeomorphic to $\R^n$. {\bf \red Then there exists a partition of unity subordinate to $\{U_\alpha\}$.} \newpage {\bf \blue Whitney's theorem for compact manifolds} \theorem Let $M$ be a compact smooth manifold. {\bf \red Then $M$ admits a closed smooth embedding to $\R^N$.} {\bf\green Proof. Step 1:} Choose a finite atlas $\{V_i, \phi_i:\; V_i \arrow \R^n, i=1, 2, ..., m\}$, and subordinate partution of unity $\mu_i:\; M \arrow [0,1]$. Let $\alpha:\; [0,1]\arrow [0,1]$ be a smooth, monotonous function mapping 0 to 0 and $[1/2m,1]$ to 1, and $\nu_i:=\alpha(\mu_i)$. {\bf\green Step 2:} Denote by $W_i$ the set of interior points of $\bar W_i:=\{z \ \ |\ \ \nu_i(z)=1\}$. Since $\sum_{i=1}^m \mu_i =1$, the set $\{W_i\}$ is a cover of $M$. {\bf\green Step 3:} For each $i$, the map $\Phi_i(z):= \left(\nu_i\phi_i(z), \sqrt{1-|\phi_i(z)|^2\nu_i(z)^2}\right)$ is injective on $W_i$ and maps $M$ to a sphere $S^n \subset \R^{n+1}$. {\bf\green Step 4:} The product map \[ \Psi:=\prod_{i=1}^m:\;\Phi_i:\; M \arrow \underbrace{S^n\times S^n \times ...\times S^n}_{\text{$m$ times}} \] is an injective, continuous map from a compact, hence it is a homeomorphism to its image {\bf \purple (prove it)}. \newpage {\bf \blue Whitney's theorem for compact manifolds (cont.)} {\bf\green Step 5:} Any smooth function on $\bar W_i:=\{z \ \ |\ \ \nu_i(z)=1 \}$ can be obtained as a restriction of a smooth function on $\Phi_i(\bar W_i)\subset S^n$, hence {\bf \purple this map induces an isomorphism of the ring of smooth functions on $\bar W_i$ and its image.} {\bf\green Step 6:} Let $W_i$ be the set of interior points of $\bar W_i$. We have constructed a cover $\{W_i\}$ of $M$ such that on each $W_i$ the map $\Psi$ induced an isomorphism of ringed spaces. \endproof \end{document}