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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 9: Serre-Swan theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf Math in Moscow and HSE \\[2mm] April 15, 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 total space of 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 $y\in Y$. \definition {\bf \blue A vector bundle} is a locally free sheaf of $C^\infty M$-modules. \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}. \remark {\bf \red This construction is an ``equivalence of categories'';} see below for a definition. \newpage {\bf \blue Categories} \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 $\Id_A \in \Mor(A,A)$. {\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 $\Id_x\circ \phi = \phi = \phi\circ \Id_Y$. \newpage {\bf \blue Functors and equivalence of categories} \definition Let $\cac_1, \cac_2$ be categories. {\bf\blue A covariant functor} from $\cac_1$ to $\cac_2$ is the following collection of data.\\ \phantom{XX}(i) A map $F:\; \Ob(\cac_1) \arrow \Ob(\cac_2)$.\\ \phantom{XX}(ii) A map $F:\; \Mor(X,Y) \arrow \Mor(F(X), F(Y))$,\\ defined for each $X, Y \in \Ob(\cac_1)$.\\ These data define {\bf\blue a functor from $\cac_1$ to $\cac_2$}, if $F(\phi) \circ F(\psi) = F(\phi\circ\psi)$, and $F(\Id_X) = \Id_{F(X)}$. \definition Two functors $F, G:\;\cac_1\arrow \cac_2$ are called {\bf\blue equivalent} if for each $X \in \Ob(\cac_1)$ there exists an isomorphism $\Psi_X:\; F(X) \arrow G(X)$, such that for each $\phi\in \Mor(X,Y)$ one has $F(\phi) \circ \Psi_Y= \Psi_X\circ G(\phi).$ \definition A functor $F:\; \cac_1 \arrow \cac_2$ is called {\bf\blue equivalence of categories} if there exist functors $G, G':\; \cac_2 \arrow \cac_1$ such that $F\circ G$ is equivalent to an identity functor on $\cac_1$, and $G' \circ F$ is equivalent to identity functor on $\cac_2$. \example Let $\cac$ be a category of finite-dimensional vector spaces ovet $\R$ with a fixed basis (morphisms are linear maps), and $\cac'$ a category with $\Ob(\cac')=\{\emptyset, \R, \R^2, \R^3, ...\}$, and morphisms also linear maps. Prove that the inclusion map $\cac' \arrow \cac$ {\bf \red is an equivalence of categories,} but {\bf \purple not an isomorphism}. \newpage {\bf \blue Total space of a vector bundle from its sheaf of sections} \definition {\bf\blue Category of vector bundles} $\cac_b$ is a category where objects are locally free $C^\infty M$-sheaves, and morphisms are morphisms of $C^\infty M$-sheaves such that all kernels and cokernels are locally free. \exercise {\bf \purple Prove that it is a category.} \definition {\bf\blue Category of total spaces of vector bundles} $\cac_t$ is a category where objects are total spaces of vector bundles, and morphisms of total spaces over $M$ are maps $B_1 \arrow B_2$ compatible with projection to $M$, the multiplicative structure, and of constant rank at each fiber. \exercise {\bf \purple Prove that it is a category.} \theorem 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 $\pi^{-1}(U)\stackrel \pi \arrow U$. Then this map {\bf \red defines an equivalence of categories $\cac_b\tilde \arrow \cac_t$.} \remark {\bf \purple The proof was given in the last lecture,} using different language. \exercise {\bf \purple Produce a proof of this theorem.} \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, and such that the quotient ${\cal B}/{\cal B}_1$ 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.} \proposition Let $A\subset B$ be a sub-bundle. {\bf \red Then $B\cong A\oplus C$.} \proof {\bf \purple Find a positive definite metric on $B$,} and set $C:=B^\bot$. \endproof \newpage {\bf \blue Tangent bundle} \proposition Let $M\subset \R^n$ be a smooth submanifold of $\R^n$, and $TM\subset \R^n \times \R^n$ the set of all pairs $(v, x) \in M \times \R^n$, where $x\in M \times \R^n$ is a point of $M$, and $v\in \R^n$ a vector tangent to $M$ in $m$, that is, satisfying \[ \lim_{t \arrow 0}\frac{d(M, m+tv)}{t} \arrow 0. \] Then the natural additive operation on $TM\subset M \times \R^n$ (addition of the second argument) and a multiplication by real numbers {\bf \blue defines on $TM$ a structure of a relative vector space over $M$}, that is, makes $TM$ a total space of a vector bundle. Moreover, this vector bundle is isomorphic to a tangent bundle, that is, to the sheaf $\Der_\R (C^\infty M)$. {\bf \green Proof. Step 1:} For each $z\in M$, we can choose coordinates in a neighbourhood of $z$ in $\R^n$ in such a way that $M=\R^k \subset \R^n$. Therefore, {\bf \purple it would suffice to prove proposition when $M=\R^k\subset \R^n$.} {\bf \green Proof. Step 2:} In this case, $TM=\R^k\times \R^k$ {\bf \purple is a total space of a vector bundle,} of the same dimension as the tangent bundle. It remains to construct a sheaf morphism from the sheaf of sections of $TM$ to $\Der_\R (C^\infty M)$, inducing an isomorphism. \newpage {\bf \blue Tangent bundle (cont.)} {\bf \green Proof. Step 3:} Let $\pi_x:\; \R^n \arrow T_x M$ be an orthogonal projection map. By the inverse function theorem, {\bf \purple $\pi_x\restrict M:\; M \arrow T_x M$ is a diffeomorphism in a neighbourhood of $x\in M$.} Let $U_x \subset T_x M$ be such an open neighbourhood and $\pi_x^{-1}(U_x)\stackrel{\pi_x}\arrow U_x$ a diffeomphism. {\bf \green Proof. Step 4:} For each vector $v\in T_xM$, and $f\in C^\infty M$, let $D_v(f)$ be the derivative of $\tilde f\in C^\infty U_x$ along $v$, where $\tilde f(z)= f(\pi_x^{-1}(z))$. Then a section $\gamma \in TM(U)$ defines a derivation $D_\gamma(f)(z):= D_{\gamma\restrict z}(f)$. {\bf \purple We obtained a sheaf homomorphism $TM\stackrel \Psi \arrow \Der_\R (C^\infty M)$.} {\bf \green Proof. Step 5:} The vector bundles $TM$ and $\Der_\R (C^\infty M)$ have the same dimension, and for each non-zero vector $v\in T_xM$, {\bf \purple the corresponding derivation is non-zero, hence $\ker \Psi=0$.} \endproof \definition The tangent bundle of $M$, as well as its total space, is denoted by $TM$. When one wants to distinguish the total space and the tangent bundle, one writes $\Tot(TM)$. \newpage {\bf \blue Pullback} \claim Let $M_1\stackrel \phi \arrow M$ be a smooth map of manifolds, and $B \stackrel \pi \arrow M$ a total space of a vector bundle. {\bf \red Then $B \times_M M_1$ is a total space of a vector bundle on $M_1$. } {\bf\green Proof. Step 1:} {\bf \purple $B \times_M M_1$ is obviously a relative vector space.} Indeed, the fibers of projection $\pi_1:\; B \times_M M_1 \arrow M_1$ are vector spaces, $\pi_1^{-1}(m_1) = \pi^{-1}(\phi(m_1))$. It remains only to show that it is locally trivial. {\bf\green Step 2:} Consider an open set $U\subset M$ that $B\restrict U = U \times \R^n$, and let $U_1:= \phi^{-1} U$. Then $B \times_U U_1= U_1 \times \R^n$. {\bf \purple Since $M_1$ is covered by such $U_1$, this implies that $\pi_1$ is a locally trivial fibration,} and the additive structure smoothly depends on $m_1\in M_1$. \endproof \definition The bundle $\pi_1:\; B \times_M M_1 \arrow M_1$ is denoted $\phi^* B$, and called {\bf\blue inverse image}, or {\bf \blue a pullback} of $B$. \newpage {\bf \blue Pullback and the tangent bundle} \claim Let $j:\; M\hookrightarrow N$ be a closed embedding of smooth bundles. {\bf \red Then there is a natural injective morphism of vector bundles $TM\hookrightarrow j^* TN$.} \proof Using Whitney's theorem, we embed $N$ to $\R^n$. Then $j^* TN\subset M\times \R^n$ is the set of pairs $x\in M, v\in T_x N$. {\bf \purple The bundle $TM$ is embedded to $j^*TN$, because each tangent vector to $M$ is also tangent to $N$.} \endproof \exercise Prove that the map $TM\hookrightarrow j^* TN$ {\bf \red is independent from the choice of embedding $N \subset \R^n$.} \corollary Let $M$ be a manifold, and $j:\; M\hookrightarrow \R^n$ a closed embedding. {\bf \red Then $TM$ is a direct summand of a trivial bundle $j^* T\R^n$.} \newpage {\bf \blue Any bundle is a direct summand of a trivial bundle} \theorem {\bf \red Any vector bundle on a metrizable manifold is a direct summand of a trivial bundle.} {\bf \green Proof. Step 1:} Let $B$ be a vector bundle on $M$, and $\Tot B$ its total space. Consider the tangent bundle $T\Tot B$, and let $M\stackrel \phi \hookrightarrow \Tot B$ be an embedding corresponding to a zero section. {\bf \purple Then the pullback $\phi^* T\Tot B$ is isomorphic (as a bundle) to the direct sum $TM \oplus B$.} {\bf \green Step 2:} Using Whitney's theorem, find a closed embedding $j:\; \Tot B\arrow \R^n$. {\bf \purple This gives injective morphisms of vector bundles \[ B \hookrightarrow TM \oplus B = \phi^*(T\Tot B)\hookrightarrow (\phi j)^*T\R^n.\]} However, $(\phi j)^*T\R^n$ is trivial, because the bundle $T\R^n$ is trivial. \endproof \newpage {\bf \blue Projective modules} \definition Let $V$ be an $R$-module, and $V'\subset V$ its submodule. Assume that $V$ contains a submodule $V''$, not intersecting $V'$, such that $V'$ together with $V''$ generate $V$. In this case, $V'$ and $V''$ are called {\bf \blue direct summands} of $V$, and $V$ -- {\bf\blue a direct sum} of $V'$ and $V''$. This is denoted $V=V' \oplus V''$. \definition An $R$-module is called {\bf\blue projective} if it is a direct summand of a free module $\bigoplus_I R$ (possibly of infinite rank). \corollary Let ${\cal B}$ be a vector bundle, and $B$ its space of sections, considered as a $C^\infty M$-module. {\bf \red Then $B$ is projective.} \theorem {\bf \blue (Serre-Swan theorem)} Let $\cac_p$ be a category with objects projective $C^\infty M$-modules, and morphisms homomorphism of $C^\infty M$-modules with kernels and cokernels projective, $\cac_b$ the category of vector bundles, and $\Psi:\; \cac_b \arrow \cac_b$ a functor mapping $B$ to its space of global sections. {\bf \red Then $\Psi$ is an equivalence of categories.} Proof later. \newpage {\bf \blue Determinant bundle} \definition {\bf \blue A line bundle} is a 1-dimensional vector bundle. \exercise Let $M$ be a simply connected manifold. {\bf \purple Prove that any real line bundle on $M$ is trivial.} \definition Let $B$ be a vector bundle of rank $n$, and $\Lambda^n B$ its top exterior product. This bundle is called {\bf\blue determinant bundle} of $B$. \remark It is a line bundle. \remark Let $M$ be an $n$-manifold, and $\Lambda^n TM$ a determinant bundle of its tangent bundle. Prove that {\bf \purple $\Lambda^n TM$ is trivial if and only if $M$ is orientable.} \definition A real vector bundle is called {\bf\blue orientable} if its determinant bundle is trivial. \newpage {\bf \blue Trivializations and determinant} \definition Recall that {\bf\blue a trivialization} of a vector bundle $B$ over $U$ is a set of {\bf \blue free generators} of $B$, that is, sections $x_1,..., x_n\in B$ such that the map $\nu:\; (C^\infty U)^n \arrow B\restrict U$ mapping generators $e_i \in (C^\infty U)^n$ to $x_i$ is an isomorphism. \definition Let $x\in M$ be a point on a manifold. Denote by ${\goth m}_x\subset C^\infty M$ the ideal of all functions vanishing in $x$. Let ${\cal B}$ be a sheaf of $C^\infty M$-modules, and $b$ a section of ${\cal B}$. We say that $b$ {\bf\blue nowhere vanishes} on $U\subset M$ if its germ $b_x$ does not lie in ${\goth m}_x {\cal B}$ for each $x\in U$. \proposition Let $B$ be a vector bundle, and $x_1,..., x_n\in B$ be a set of sections which are linearly independent in $B/{\goth m}_{z_0}B$ and generate $B/{\goth m}_{z_0}B$, for a fixed point $z_0\in M$. Let $\xi\in \Lambda^n B$, $\xi:= x_1 \wedge x_2 \wedge ... \wedge x_n$ be the determinant of $x_i$, considered as a section of a line bundle $\det B$. Suppose that $\xi$ nowhere vanishes on $U\subset M$. {\bf \purple Then $\left\{x_i\restrict U\right\}$ are free generators of $B\restrict U$.} \proof Define a map $\nu:\; (C^\infty U)^n \arrow B\restrict U$ mapping generators $e_i \in (C^\infty U)^n$ to $x_i$. {\bf \purple This map induces an isomorphism on each fiber, hence bijective. } The inverse function theorem implies that it is a diffeomorphism. \endproof \newpage {\bf \blue A stalk of a $C^\infty M$-module} \definition Let $x\in M$ be a point on a manifold. {\bf\blue A stalk} of a $C^\infty M$-module $V$ is a tensor product $C^\infty_x M\otimes_{C^\infty M} V$, where $C^\infty_x M$ is a ring of germs of $C^\infty M$ in $x$. We consider a stalk $V_x$ as a $C^\infty_x M$-module. \remark Let $V$ be a free $C^\infty M$-module. Then stalk of the space of sections $V(M)$ in $x$ is a stalk of the sheaf $V$ in $x$. \claim Let $A$ be a free $C^\infty M$-module of rank $n$, decomposed as a direct sum of two projective modules: $A= B\oplus C$. We identify $A$ with a space of sections of a trivial sheaf of $C^\infty M$-modules, denoted by ${\cal A}$. {\bf \blue Let ${\cal B}\subset {\cal A}$ be a subsheaf consisting of all sections $\gamma \in {\cal V}(U)$, such that the germs of $\gamma$ at each $x\in M$ lie in the stalk $B_x$.} Define ${\cal C}\subset {\cal A}$ in a similar fashion. Then\\ \phantom{xx} (i) ${\cal B}$, ${\cal C}$ are sheaves of $C^\infty M$-modules.\\ \phantom{xx} (ii) ${\cal A} ={\cal B}\oplus {\cal C}$.\\ \phantom{xx} (iii) {\bf \red The sheaves ${\cal B}$, ${\cal C}$ are locally free.} \proof Next slide. \newpage {\bf \blue The proof of Serre-Swan theorem} \claim Let $A$ be a free $C^\infty M$-module of rank $n$, decomposed as a direct sum of two projective modules: $A= B\oplus C$. We identify $A$ with a space of sections of a trivial sheaf of $C^\infty M$-modules, denoted by ${\cal A}$. {\bf \blue Let ${\cal B}\subset {\cal A}$ be a subsheaf consisting of all sections $\gamma \in {\cal V}(U)$, such that the germs of $\gamma$ at each $x\in M$ lie in the stalk $B_x$.} Define ${\cal C}\subset {\cal A}$ in a similar fashion. Then\\ \phantom{xx} (i) ${\cal B}$, ${\cal C}$ are sheaves of $C^\infty M$-modules.\\ \phantom{xx} (ii) ${\cal A} ={\cal B}\oplus {\cal C}$.\\ \phantom{xx} (iii) {\bf \red The sheaves ${\cal B}$, ${\cal C}$ are locally free.} \proof The first two claims are clear. Fix $z\in M$. Let $x_1, ..., x_k$ be sections of ${\cal B}$ generating ${\cal B}/{\goth m}_z {\cal B}$ and $y_1, ..., y_l$ sections of ${\cal C}$ generating ${\cal C}/{\goth m}_z {\cal C}$. Choose them to be linearly independent, and let $U$ be an open neighbourhood of $z$ such that the section $x_1\wedge x_2 \wedge ... \wedge x_k \wedge y_1\wedge ... \wedge y_l\in \Lambda^n B$ is nowhere degenerate on $U$. Then $\{x_i, y_j\}$ are free generators of ${\cal A}$, hence $\{x_i\}$ are free generators of ${\cal B}$ and $\{y_j\}$ are free generators of ${\cal C}$. {\bf \red We have shown that these sheaves are locally free}. \endproof \remark This gives a way of reconstructing a vector bundle from a projective $C^\infty M$-module. {\bf \purple The rest of the proof of Serre-Swan is left as an exercise.} \end{document}