%\newpage % %{\bf \blue \'Etal\'e space of a sheaf} % %\definition %Let ${\cal F}$ be a sheaf on $M$, and $E({\cal F}):= \bigcup_{x\in M}{\cal F}_x$ %the union of all stalks of ${\cal F}$. Given a section %$f\in {\cal F}(U)$, let $U\arrow E({\cal F})$ %map each $x\in U$ to the germ of $f$ in $x$. %The image of $f$ under this map is called $e(f)$. %We equip $E({\cal F})$ %with a weakest topology for which each $e(f)$ is open. %The space $E({\cal F})$ is called {\bf \blue \'etal\'e %space of a sheaf}. % %\remark %The space $E({\cal F})$ is usually non-Hausdorff; however, %the projection $E({\cal F})\arrow M$ is locally %a diffeomorphism. The sheaf ${\cal F}$ can be reconstructed % %\exercise % \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{\Der}{\operatorname{Der}} \newcommand{\even}{{\rm even}} \newcommand{\ev}{{\rm even}} \newcommand{\odd}{{\rm odd}} \newcommand{\const}{{\it const}} \newcommand{\diam}{{\sf diam}} \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{\Sup}{\operatorname{Sup}} \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 7 \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 7: Categories and locally trivial fibrations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf Math in Moscow and HSE \\[2mm] April 1, 2013 } \end{center} \newpage {\bf \blue Locally trivial fibrations} \definition A smooth map $f:\; X \arrow Y$ is called {\bf \blue a locally trivial fibration} if each point $y\in Y$ has a neighbourhood $U\ni y$ such that $f^{-1}(U)$ is diffeomorphic to $U\times F$, and the map $f:\; f^{-1}(U)= U\times F \arrow U$ is a projection. In such situation, $F$ is called {\bf \blue the fiber} of a locally trivial fibration. \definition {\bf \blue A trivial fibration} is a map $X \times Y \arrow Y$. \example The projection $S^3\subset \C^2\backslash 0 \stackrel f\arrow \C P^1$ is called {\bf \blue the Hopf fibration}. Given $U= \{x:1\}\subset \C P^1$, with $|x|\leq 1$, one has \[ f^{-1}(U)= \{z_1, z_2\in S^3 \ \ | \ \ |z_1|^2 +|z_2|^2=1, |z_1|\leq 1\} \] (here $z_i$ are complex coordinates in $\C^2$). Then \[ f^{-1}(U)= \left\{ (z_1, z_2) \ \ |\ \ z_2\in U(1)\cdot \sqrt{1-|z_1|^2} \right\}, \] where $U(1)=\{z\in \C \ \ |\ \ |z|=1\}$. Therefore, {\bf \red the Hopf fibration $f:\; S^3 \arrow S^2$ is a locally trivial fibration}. \remark Since $\pi_1(S^3)=0$ and $\pi_1(S^1 \times S^2)=\Z$, {\bf \red the Hopf fibration is non-trivial.} \newpage {\bf \blue Vector bundles} \definition {\bf \blue A vector bundle} on $Y$ is a locally trivial fibration $f:\; X\arrow Y$ with fiber $\R^n$, with each fiber $V:= f^{-1}(y)$ equipped with a structure of a vector space, smoothly depending on $y\in Y$. \remark This definition {\bf \green is not very precise or rigorous,} because ``smoothly depending on $y\in Y$'' {\bf \red needs to be explained.} \remark This definition is compatible with the one we used previously (``a vector bundle is a locally free sheaf of $C^\infty M$-modules''). This will be explained later. For a more rigorous approach: 1. Define categories. 2. Define group objects and vector space objects 3. Formulate ``smoothly depending on $y\in Y$'' in these terms. \newpage {\bf \blue Categories: data} \newcommand{\Ob}{\operatorname{{\cal O}b}} \newcommand{\Mor}{\operatorname{{\cal M}or}} \definition {\bf \blue A category} $\cac$ is a collection of data called ``objects'' and ``morphisms between objects'' which satisfies the axioms below. {\bf \green DATA.}\\ \phantom{XX} {\bf \red Objects:} The set $\Ob(\cac)$ of {\bf \blue objects} of $\cac$. \\ \phantom{XX} {\bf \red Morphisms:} For each $X, Y\in \Ob(\cac)$, one has a set $\Mor(X,Y)$ of {\bf \blue morphisms from $X$ to $Y$}.\\ \phantom{XX} {\bf \red Composition of morphisms:} For each $\phi\in \Mor(X,Y), \psi \in \Mor(Y,Z)$ there exists {\bf \blue the composition} $\phi\circ \psi \in \Mor(X, Z)$\\ \phantom{XX} {\bf \red Identity morphism:} For each $A\in \Ob(\cac)$ there exists a morphism $\Id_A \in \Mor(A,A)$. \remark In some versions of axiomatic set theory, one considers not a set, but {\bf \purple a class} of objects, which could be arbitrarily big, such as the class of all sets, or the class of all linear spaces. The category with {\bf \red a set} of morphisms and objects is called {\bf \blue a small category}, and one with a class {\bf \blue a big category}. In ZFC, one postulates existence of so-called {\bf \blue Grothendieck universe} (that is, {\bf \blue a strongly inaccessible cardinal}). {\bf \blue Small sets} are ones which belong to the Grothendieck universe, the rest of the sets are {\bf \blue big}. {\small Existence of a strongly inaccessible cardinal {\bf \green implies consistency of ZFC} (Goedel).} \newpage {\bf \blue Categories: axioms} {\bf \green AXIOMS.}\\ \phantom{XX} {\bf \red Associativity of composition:} $\phi_1\circ(\phi_2\circ\phi_3)=(\phi_1\circ\phi_2)\circ\phi_3$.\\ \phantom{XX} {\bf \red Properties of identity morphism:} For each $\phi\in \Mor(X,Y)$, one has $\Id_x\circ \phi = \phi = \phi\circ \Id_Y$\\ \definition Let $X, Y\in \Ob(\cac)$ -- objects of $\cac$. A morphism $\phi\in \Mor(X,Y)$ is called {\bf \blue an isomorphism} if there exists $\psi\in \Mor(Y,X)$ such that $\phi \circ \psi = \Id_X$ and $\psi\circ\phi = \Id_Y$. In this case, the objects $X$ and $Y$ are called {\bf \blue isomorphic}. {\bf \green Examples of categories:}\\ {\bf \purple Category of sets:} its morphisms are arbitrary maps.\\ {\bf \purple Category of vector spaces:} its morphisms are linear maps.\\ {\bf \purple Categories of rings, groups, fields:} morphisms are homomorphisms.\\ {\bf \purple Category of topological spaces:} morphisms are continuous maps.\\ {\bf \purple Category of smooth manifolds:} morphisms are smooth maps. {\bf \red It is often convenient to express morphisms by arrows, and call them ``maps''.} \newpage {\bf \blue Some categorical constructions} \definition {\bf \blue A product} $X_1\times X_2$ of $X_1, X_2 \in \Ob(cac)$ is an object of $\cac$ equipped with {\bf \blue projection maps} $\pi_i:\; X_1\times X_2\arrow X_i$ such that {\bf \purple for any pair of morphisms $\phi_i\in \Mor(Y, X_i)$ there is a unique morphism $\phi\in \Mor(Y, X_1 \times X_2)$ such that $\phi\circ\pi_i =\phi_i$.} \exercise Prove that {\bf \purple a product is unique up to isomorphism,} if it exists. \exercise Prove that the product is {\bf \red associative:} $X\times(Y\times Z)\cong (X\times Y)\times Z$ and {\bf \red commutative:} $X\times Y\cong Y\times X$. \exercise Find the product in the categories of a. groups b. rings c. vector spaces d. sets e. topological spaces. \definition {\bf\blue An initial object} of a category is an object $I\in \Ob(\cac)$ such that $\Mor(I,X)$ is always a set of one element. {\bf \blue A terminal object} is $T\in \Ob(\cac)$ such that $\Mor(X,T)$ is always a set of one element. \exercise Prove that {\bf \purple the initial and the terminal object is unique,} up to isomorphism. %\exercise Find the initial and the terminal object in the categories of %a. groups b. rings c. vector spaces d. sets e. topological spaces, %or show that it does nor exist. \newpage {\bf \blue Group objects in categories} \exercise Let $T$ be a terminal object. {\bf \purple Prove that $X\times T\cong X$ for each $X\in \Ob(\cac)$.} \definition An object $G\in \Ob(\cac)$ is called {\bf \blue a group object} if there exists a morphism $\mu\in \Mor(G\times G, G)$ ({\bf \blue the product}), a morphism $e\in \Mor(T, G)$ from the terminal object ({\bf \blue the unit}), and a morphism $i\in \Mor(G,G)$ ({\bf \blue the inverse}), satisfying the following axioms. {\bf \red Associativity:} Consider the morphisms $\mu_{12},\mu_{23}:\; G\times G\times G\arrow G\times G$, the first map takes the product on the first two objects, and acts as identity on the third, the second maps is a product on last 2 and identity on the first. Then $\mu_{12}\circ\mu=\mu_{23}\circ \mu:\; G\times G\times G\arrow G$. {\bf \red Unit:} The compositions $G=G\times T \stackrel{\Id_G\times e}\arrow G\times G \stackrel \mu \arrow G$ and $G=G\times T \stackrel{e\times \Id_G}\arrow G\times G \stackrel \mu \arrow G$ are identities. {\bf \red Inverse:} Let $\Delta:\; G \arrow G\times G$ be {\bf \blue the diagonal map,} that is, a map $G \arrow G\times G$ obtained from a pair of identity maps. Then the composition $G\stackrel\Delta\arrow G\times G\stackrel{\Id_G\times i}\arrow G\times G \stackrel \mu \arrow G$ is equal to $G\arrow T \stackrel e\arrow G$. \newpage {\bf \blue Examples of group objects} \example {\bf \blue A topological group} is a group object in the category of topological spaces. \example {\bf \blue A Lie group} is a group object in the category of smooth manifolds. \definition Let $\cac$ be a category. {\bf \blue An opposite category} $\cac^\circ$ is a category with the same sets of objects, $\Mor_{\cac^o}(X,Y)=\Mor_\cac(Y,X)$, with the same compositions as in $\cac$ taken in inverse order. \example {\bf \red The category of finitely generated algebras withous nilpotents over $\C$ is equivalent to $\cac^\circ$,} where $\cac$ is a category with objects algebraic subsets in $\C^n$ (common zeros of a system of polynomial equations) and morphisms polynomial functions. This statement is called {\bf \blue ``Hilbert's Nullstellensatz''.} \example {\bf \blue An algebraic group} is a group object in the category $\cac^\circ$, where $\cac$ is a category of rings. \example {\bf \blue A formal group} is a group object in the category $\cac^o$, where $\cac$ is a category of complete local rings (over $\C$, these are local rings, obtained as quotients of the ring of formal power series by an ideal). \newpage {\bf \blue Topological groups over a base} \definition Fix a topological space $M$, and let $\cac_M$ be a category of pairs $(X, f:\; X\arrow M)$ with morphisms being continuous maps from $X_1$ to $X_2$ commuting with the projections to $M$. The product in $\cac_M$ is called {\bf \blue fiber product:} $X_1\times_M X_2:= \{(x_1,x_2)\in X_1\times X_2\ \ |\ \ f_1(x_1)=f_2(x_2)\}$. A group object in $\cac_M$ is called {\bf \blue a topological group over $M$}. \remark {\bf \red This definition is equivalent to the following.} \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\blue 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. \\ \phantom{X} A section $M \stackrel e \arrow B$ is called {\bf\blue 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 $Id_B$. \\ \phantom{X} A morphism $\nu:\; B \arrow B$ over $M$ is called {\bf \blue a 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\blue a topological group over $M$}. \newpage {\bf \blue Vector spaces over a base} \remark Let $\pi:\; G \arrow M$ be a topological group over $M$. Then the fiber $\pi^{-1}(m)$ is a group for each $m\in M$. {\bf \red This group structure depends on $m\in M$ continuously}, but {\bf \purple to state this dependency formaly, one needs to define a topological group over $M$.} \definition Let $k$ be a field. {\bf\blue A $k$-vector space object} in a category $\cac$ is a group object $V$ equipped with a set of morphisms $\lambda_x\in \Mor(V,V)$, parametrized by $x\in k$, and satisfying the following conditions.\\ \phantom{X} {\bf \red Multiplicativity:} $\lambda_x \lambda_y=\lambda_{xy}$, \\ \phantom{X} {\bf \red Zero:} $\lambda_1 =\Id_V$\\ \phantom{X} {\bf \red Unit:} $\lambda_0:\; V \arrow V$ is a composition $V \arrow T \stackrel e \arrow V$.\\ \phantom{X} {\bf \red Additivity:} Let $\Delta$ be the diagonal map. Then the composition $G\stackrel\Delta \arrow G\times G \arrow \lambda_x\times\lambda_y\stackrel\mu\arrow G$ is equal to $\lambda_{x+y}$.\\ \phantom{X} {\bf \red Distributivity:} The composition $G\times G \stackrel{\lambda_x\times \lambda_x}\arrow G\times G \stackrel \mu\arrow G$ is equal to $\mu\circ \lambda_x$. \definition Let $k$ be a topological field (for instance, $\C$ or $\R$). {\bf\blue A topological vector space $B$ over a base $M$} is a vector space object in $\cac_M$, such that the map $\lambda_x:\; k\times B \arrow B$ is continuous. \newpage {\bf \blue Vector spaces over a base (category-free definition)} \definition 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 $\phi_{\lambda+\lambda'}(g)= \phi_{\lambda}(g)+\phi_{\lambda'}(g).$ Then $G$ is called {\bf \blue a vector space over $k$}. \definition Let $k=\R$ or $\C$. An abelian topological group $B\stackrel \pi \arrow M$ over $M$ is called {\bf \blue a vector space over a base $M$}, or {\bf \blue a 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 the above definition. \remark 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 $\pi^{-1}(U) \stackrel \pi \arrow U$. Then {\bf \red ${\cal B}(U)$ defines a sheaf of modules over a sheaf $C^0(M)$ of continuous functions.} \example 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\blue tangent cone} $C_sS$ if the distance from $S$ to a point $s+tv$ converges to 0 as $t\rightarrow 0$ faster than linearly: $\lim\limits_{t \rightarrow 0}\frac{d(S, s+tv)}{t} = 0.$ {\bf \purple Then the set $CS$ of all pairs $(s, v), s\in S, v\in C_s S$ is a relative vector space over $S$}. \newpage {\bf \blue Total space of a vector bundle} \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\blue a total space of a vector bundle.} \remark Let $\pi:\; B \arrow M$ be a total space of a vector bundle, $U\subset M$ open subset, and ${\cal B}(U)$ the space of all smooth sections of $\pi^{-1}(U)\stackrel \pi \arrow U$. {\bf \purple Then ${\cal B}$ is a locally free sheaf of $C^\infty M$-modules}. \theorem {\bf \red Every locally free sheaf $C^\infty M$-modules is defined from a total space of a vector bundle,} which is determined uniquely by a sheaf. The proof will be a couple of slides below. \newpage {\bf \blue Fiber of a locally free sheaf} \definition Let ${\cal B}$ be an $n$-dimensional locally free sheaf of $C^\infty$-modules 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\blue the fiber} of ${\cal B}$ in $x$ as a quotient ${\cal B}_x/{\goth m}_x{\cal B}_x$. A fiber of ${\cal B}$ is denoted ${\cal B}\restrict x$. \remark {\bf \purple A fiber of an $n$-dimensional bundle is an $n$-dimensional vector space.} \remark Let ${\cal B}= C^\infty M^n$, 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 $\Psi$ from the set of all fibers ${\cal B}$ to $M \times \R^n$, mapping $(x, \phi=(f_1, ..., f_n))$ to $(f_1(x), ..., f_n(x))$. {\bf \red Then $\Psi$ is bijective.} Indeed, ${\cal B}\restrict x=\R^n$. \newpage {\bf \blue Total space of a vector bundle from its sheaf of sections} \definition Let ${\cal B}$ be an $n$-dimensional locally free sheaf of $C^\infty$-modules. 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 $\Tot {\cal B}(U)=U \times \R^n$ 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 ${\cal B}\restrict U$. Then $\Tot {\cal B}$ is called {\bf \blue a total space of a vector bundle ${\cal B}$.} \claim The space $\Tot {\cal B}$ with this topology {\bf \purple is a locally trivial fibration over $M$, with fiber $\R^n$.} Moreover, it is a relative vector space over $M$, and {\bf \purple the sheaf of smooth sections of $\Tot {\cal B}\arrow M$ is isomorphic to ${\cal B}$.} \remark {\bf \red This gives an equivalence between locally free sheaves of $\C^\infty$-modules and the total spaces of vector bundles,} defined abstractly in terms of locally trivial fibrations. \end{document}