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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 8: Vector bundles 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 $X \times Y \arrow Y$. \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 {\bf \purple 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 today. \newpage {\bf \blue Fiber product} \definition Fix a topological space $M$, and let $\pi_1:\; X_1 \arrow M$ and $\pi_2:\; X_2 \arrow M$ be continuous maps. The {\bf \blue fiber product} is defined as \[ X_1\times_M X_2:= \{(x_1,x_2)\in X_1\times X_2\ \ |\ \ \pi_1(x_1)=\pi_2(x_2)\}. \] \remark Consider the projection $X_1\times_M X_2\stackrel \pi\arrow M$. Then \[ \pi^{-1}(m)= \pi_1^{-1}(m)\times \pi_2^{-1}(m). \] \exercise Let $\pi_i:\; X_i \arrow M$, $i=1,2$ be trivial fibrations, $X_i= M\times F_i$. {\bf \purple Prove that $X_1 \times_M X_2 = M\times F_1\times F_2$.} \remark If $X_i$ are locally trivial fibrations over $M$ with fiber $F_i$, {\bf \purple the fiber product $X_1\times_M X_2$ is a locally trivial fibration over $M$ with fiber $F_1 \times F_2$.} \newpage {\bf \blue Fiber product: universal property} \remark The fiber product satisfies the following {\bf \blue universal property}. Let $X_1\times_M X_2\stackrel {\tilde \pi_i} \arrow X_i$ be the natural projection. Then any commutative square \[\begin{CD} Y@>{f_1}>> X_1\\ @V{f_2}VV @VV{\pi_1}V\\ X_2 @>{\pi_2}>> M \end{CD} \] induces a unique continuous map $f:\; Y \arrow X_1\times_M X_2$ such that $f\circ {\tilde \pi_i}=f_i$. Moreover, {\bf \red any space satisfying the universal property is homeomorphic to $X_1\times_M X_2$.} \remark This statement is awkward, because I avoided the language of categories. \exercise {\bf \purple Translate this property to the language of categories.} \newpage {\bf \blue Topological groups over a base} \definition Fix a topological space $M$. {\bf \blue A space over $M$} $(X, f:\; X \arrow M)$ is a topological space equipped with a continuous map to $M$. {\bf A morphisms} $\phi:\; (X_1, f_1) \arrow (X_2, f_2)$ is a continuous map $\phi:\; X_1 \arrow X_2$ commuting with projections to $M$. \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} \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. \newpage {\bf \blue Tensor product} \definition Let $V,V'$ be $R$-modules, $W$ a free abelian group generated by $v\otimes v'$, with $v\in V, v'\in V'$, and $W_1\subset W$ a subgroup generated by combinations $rv \otimes v'-v\otimes rv'$, $(v_1+ v_2)\otimes v'- v_1 \otimes v' - v_2 \otimes v'$ and $v\otimes (v'_1+ v'_2)- v\otimes v'_1 - v\otimes v'_2$. Define {\bf\blue the tensor product} $V \otimes_R V'$ as a quotient group $W/W_1$. \exercise Show that $r \cdot v\otimes v'\mapsto (rv)\otimes v'$ {\bf \purple defines an $R$-module structure on $V \otimes_R V'$.} \remark Let ${\cal F}$ be a sheaf of rings, and ${\cal B}_1$ and ${\cal B}_2$ be sheaves of locally free $(M, {\cal F})$-modules. {\bf \purple Then \[ U \arrow {\cal B}_1(U)\otimes_{{\cal F}(U)} {\cal B}_2(U) \] is also a locally free sheaf of modules.} \definition {\bf \blue Tensor product} of vector bundles is a tensor product of the corresponding sheaves of modules. \newpage {\bf \blue Dual bundle and bilinear forms} \definition Let $V$ be an $R$-module. {\bf\blue A dual $R$-module} $V^*$ is $\Hom_R(V, R)$ with the $R$-module structure defined as follows: $r\cdot h(\dots) \mapsto rh(\dots)$. \claim Let ${\cal B}$ be a vector bundle, that is, a locally free sheaf of $C^\infty M$-modules, and $\Tot{\cal B}\stackrel \pi \arrow M$ its total space. Define ${\cal B}^*(U)$ as a space of smooth functions on $\pi^{-1}(U)$ linear in the fibers of $\pi$. {\bf \purple Then ${\cal B}^*(U)$ is a locally free sheaf over $C^\infty(M)$.} \definition This sheaf is called {\bf \blue the dual vector bundle}, denoted by $B^*$. Its fibers are dual to the fibers of $B$. \definition {\bf \blue Bilinear form} on a bundle ${\cal B}$ is a section of $({\cal B}\otimes {\cal B})^*$. A symmetric bilinear form on a real bundle ${\cal B}$ is called {\bf\blue positive definite} if it gives a positive definite form on all fibers of ${\cal B}$. Symmetric positive definite form is also called {\bf\blue a metric}. A skew-symmetric bilinear form on ${\cal B}$ is called {\bf\blue non-degenerate} if it is non-degenerate on all fibers of ${\cal B}$. \newpage {\bf \blue Subbundles} \definition {\bf\blue A subbundle} ${\cal B}_1\subset {\cal B}$ is a subsheaf of modules which is also a vector bundle. \definition {\bf\blue Direct sum} $\oplus$ of vector bundles is a direct sum of corresponding sheaves. \example Let ${\cal B}$ be a vector bundle equipped with a metric (that is, a positive definite symmetric form), and ${\cal B}_1 \subset {\cal B}$ a subbundle. Consider a subset $\Tot {\cal B}_1^\bot\subset \Tot {\cal B}$, consisting of all $v\in {\cal B}\restrict x$ orthogonal to ${\cal B}_1\restrict x \subset {\cal B}\restrict x$. {\bf \purple Then $\Tot {\cal B}_1^\bot$ is a total space of a subbundle, denoted as ${\cal B}_1^\bot\subset {\cal B}$,} and we have an isomorphism ${\cal B}= {\cal B}_1\oplus {\cal B}_1^\bot$. \remark A total space of a direct sum of vector bundles ${\cal B}\oplus {\cal B}'$ {\bf \purple is homeomorphic to $\Tot {\cal B}\times_M \Tot {\cal B}'$.} \exercise Let ${\cal B}$ be a real vector bundle. {\bf \red Prove that ${\cal B}$ admits a metric.} \end{document} % %\newpage % %{\bf \blue Morphisms of sheaves} % %\definition %Let ${\cal B}, {\cal B}'$ be sheaves on %$M$. {\bf\blue A sheaf morphism} from ${\cal B}$ to ${\cal B}'$ %is a collection of homomorphisms ${\cal B}(U)\arrow {\cal B}'(U)$, %defined for each open subset $U\subset M$, %and compatible with the restriction maps: %\[ %\begin{CD} %{\cal B}(U) @>>> {\cal B}'(U)\\ %@VVV@VVV\\ %{\cal B}(U_1) @>>> {\cal B}'(U_1) %\end{CD} %\] % %\remark %Morphisms of sheaves of modules are defined in the same %way, but in this case {\bf \purple the maps ${\cal B}(U)\arrow {\cal % B}'(U)$ should be compatible with the module structure.} % % %\definition %A sheaf morphism is called {\bf\blue injective} %if it is injective on germs %and {\bf\blue surjective}, if it is surjective on germs. % % %\definition %Let ${\cal B} \stackrel \phi \arrow {\cal B}'$ be a %morphism of locally free sheaves of $C^\infty M$-modules. %It is called {\bf\blue a smooth morphism}, or %{\bf\blue a morphism of vector bundles} if %$\phi$ has locally free kernel and locally %free cokernel. %