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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 6: Germs and duality} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf Math in Moscow and HSE \\[2mm] March 11, 2013 } \end{center} \newpage {\bf \blue Sheaves (reminder)} \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 The definition of a sheaf below is a more abstract version of the notion of ``sheaf of functions'' defined previously. \definition A {\bf\blue presheaf} on a topological space $M$ is a collection of vector spaces ${\cal F}(U)$, for each open subset $U\subset M$, together with {\bf \blue restriction maps} $R_{UW}{\cal F}(U)\arrow {\cal F}(W)$ defined for each $W\subset U$, such that for any three open sets $W\subset V\subset U$, $\Psi_{UW}=\Psi_{UV}\circ \Psi_{VW}$. Elements of ${\cal F}(U)$ are called {\bf \blue sections of ${\cal F}$ over $U$}, and restriction map often denoted $f\restrict W$ \definition A presheaf ${\cal F}$ is called {\bf\blue a sheaf} if for any open set $U$ and any cover $U=\bigcup U_I$ the following two conditions are satisfied.\\ \phantom{xu} 1. Let $f\in {\cal F}(U)$ be a section of ${\cal F}$ on $U$ such that its restriction to each $U_i$ vanishes. {\bf \purple Then $f=0$.} \\ \phantom{xu} 2. Let $f_i\in{\cal F}(U_i)$ be a family of sections compatible on the pairwise intersections: $f_i|_{U_i\cap U_j}=f_j|_{U_i\cap U_j}$ for every pair of members of the cover. {\bf \purple Then there exists $f\in{\cal F}(U)$ such that $f_i$ is the restriction of $f$ to $U_i$ for all $i$.} \newpage {\bf \blue Sheaves and exact sequences (reminder)} \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 $\eta\restrict{U_i\cap U_j}$ and $-\eta\restrict{U_j\cap U_i}$. \newpage {\bf \blue Ringed spaces (reminder)} \definition {\bf\blue A sheaf of rings} is a sheaf ${\cal F}$ such that all the spaces ${\cal F}(U)$ are rings, and all restriction maps are ring homomorphisms. \definition {\bf \blue A sheaf of functions} is a subsheaf in a sheaf of all functions, closed under multiplication. {\bf \red For simplicity, I assume now that a sheaf of rings is a subsheaf in a sheaf of all functions}. \definition A {\bf\blue ringed space} $(M,{\cal F})$ is a topological space equipped with a sheaf of rings. 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. \newpage {\bf \blue Smooth manifolds (reminder)} \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 Partition of unity (reminder)} \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) $\sum_if_i=1$ \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 $\{U_\alpha\}$.} \definition Let $U\subset V$ be open subsets in $M$. We write $U\Subset V$ if the closure of $U$ is contained in $V$. \definition Let $f\in {\cal F}(M)$ be a section of a sheaf ${\cal F}$ on $M$. A point $x\in M$ does not lie in the {\bf\blue support} $\Sup(f)$ of $f$ if $f\restrict U=0$ for some neighbourhood $U\ni x$. \remark {\bf \purple Support of a section is obviously closed.} \newpage {\bf \blue Direct limits} \definition {\bf\blue Commutative diagram} of vector spaces is given by the following data. There is a directed graph (graph with arrows). For each vertex of this graph we have a vector space, and each arrow corresponds to a homomorphism of the associated vector spaces. {\bf \purple These homomorphism are compatible, in the following way.} Whenever there exist two ways of going from one vertex to another, the compositions of the corresponding arrows are equal. \definition Let ${\cal C}$ be a commutative diagram of vector spaces, $A, B$ -- vector spaces, corresponding to two vertices of a diagram, and $a\in A, b\in B$ elements of these vector spaces. Write $a\sim b$ if $a$ and $b$ are mapped to the same element $d\in D$ by a composition of arrows from ${\cal C}$. Let $\sim$ be an equivalence relation generated by such $a\sim b$. A quotient $\bigoplus_i C_i/E$ is called {\bf\blue a direct limit} of a diagram $\{C_i\}$. The same notion is also called {\bf\blue colimit} and {\bf\blue inductive limit}. Direct limit is denoted $\lim\limits_\rightarrow$. \definition Let ${\cal F}$ be a sheaf on $M$, $x\in M$ a point, and $\{U_i\}$ the set of all neighbourhoods of $x$. Consider a diagram with the set of vertices indexed by $\{U_i\}$, and arrows from $U_i$ to $U_j$ corresponding to inclusions $U_j \hookrightarrow U_i$. The {\bf\blue space of germs} of ${\cal F}$ in $x$ is a direct limit $\lim\limits_\rightarrow{\cal F}(U_i)$ over this diagram. The space of germs is also called {\bf \blue a stalk} of a sheaf. \newpage {\bf \blue Germs of functions} \definition A diagram ${\cal C}$ is called {\bf\blue filtered} if for any two vertices $C_i, C_j$, there exists a third vertex $C_k$, and sequences of arrows leading from $C_i$ to $C_k$ and from $C_j$ to $C_k$. \example {\bf \red The diagram formed by all neighbourhoods of a point is obviously filtered.} \claim Let ${\cal C}$ be a commutative diagram of vector spaces $C_i$, with all $C_i$ equipped with a ring structure, and all arrows ring homomorphisms. Suppose that the diagram ${\cal C}$ is filtered. {\bf \purple Then there exists a unique ring structure on $C:=\lim\limits_\rightarrow C_i$ such that all the maps $C_i \arrow C$ are ring homomorphisms.} \definition Let $M, {\cal F}$ be a ringed space, $x\in M$ its point, and $\{U_i\}$ the set of all its neighbourhoods. Consider a commutative diagram with vertices indexed by $\{U_i\}$, and arrows from $U_i$ to $U_j$ corresponding to inclusions $U_j \hookrightarrow U_i$. For each vertex $U_i$ we take a vector space of sections ${\cal F}(U_i)$, and for each arrow the corresponding restriction map. The direct limit of this diagram is called {\bf\blue the ring of germs of the sheaf ${\cal F}$ in $x$}. \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: {\small \[ \begin{CD} {\cal B}(U) @>>> {\cal B}'(U)\\ @VVV@VVV\\ {\cal B}(U_1) @>>> {\cal B}'(U_1) \end{CD} \] } \!\!\!\!\!\!\!\!\definition A sheaf morphism is called {\bf\blue injective}, or {\bf \blue a monomorphism} if it is injective on stalks and {\bf \blue surjective}, or {\bf \blue epimorphism} if it is surjective on stalks. \exercise Let ${\cal B} \stackrel \phi \arrow {\cal B}'$ be an injective morphism of sheaves on $M$. {\bf \purple Prove that $\phi$ induces an injective map ${\cal B}(U) \arrow {\cal B}'(U)$ for each $U$.} \remark A sheaf epimorphism ${\cal B} \stackrel \phi \arrow {\cal B}'$ {\bf \red does not necessarily induce a surjective map ${\cal B}(U) \arrow {\cal B}'(U)$.} \definition {\bf\blue A sheaf isomorphism} is a homomorphism $\Psi:\; {\cal F}_1 \arrow {\cal F}_2$, for which there exists an homomorphism $\Phi:\; {\cal F}_2 \arrow {\cal F}_1$, such that $\Phi\circ \Psi =\Id$ and $\Psi\circ \Phi =\Id$. \exercise Show that a morphism of sheaves $\Psi:\; {\cal F}_1 \arrow {\cal F}_2$ {\bf \red is an isomorphism if and only if it is epi and mono.} \newpage {\bf \blue Sheaves of modules} \remark Let $A:\; \phi \arrow B$ be a ring homomorphism, and $V$ a $B$-module. {\bf \purple Then $V$ is equipped with a natural $A$-module structure: $a v:= \phi(a) v$.} \definition Let ${\cal F}$ be a sheaf of rings on a topological space $M$, and ${\cal B}$ another sheaf. It is called {\bf\blue a sheaf of ${\cal F}$-modules} if for all $U\subset M$ the space of sections ${\cal B}(U)$ is equipped with a structure of ${\cal F}(U)$-module, and for all $U'\subset U$, the restriction map ${\cal B}(U) \stackrel{\phi_{U,U'}}\arrow {\cal B}(U')$ is a homomorphism of ${\cal F}(U)$-modules (use the remark above to obtain a structure of ${\cal F}(U)$-module on ${\cal B}(U')$). \definition A {\bf \blue free sheaf of modules} ${\cal F}^n$ over a ring sheaf ${\cal F}$ maps an open set $U$ to the space ${\cal F}(U)^n$. \definition {\bf\blue Locally free sheaf of modules} over a sheaf of rings ${\cal F}$ is a sheaf of modules ${\cal B}$ satisfying the following condition. For each $x\in M$ there exists a neighbourhood $U\ni x$ such that the restriction ${\cal B}\restrict U$ is free. \definition {\bf\blue A vector bundle} on a smooth manifold $M$ is a locally free sheaf of $C^\infty M$-modules. \newpage {\bf \blue Dual sheaves} \claim Let $U\subset V$ be open subsets of a Hausdorff space $M$. {\bf \red A section $s\in {\cal F}(U)$ with compact support $Z\subset U$ can be uniquely extended to $\tilde s\in {\cal F}(U)$, also with support in $Z$.} {\bf \green Proof:} $Z$ is a closed subset of $U$, not intersecting $M\backslash V$. Let $U_1$ be an open neighbourhood of $Z$ not intersecting $M\backslash V$, and $U_2:=M \backslash Z$. Then $\{U_1, U_2\}$ is a cover of $M$, and $s \restrict {U_1\cap U_2}=0$, hence {\bf \purple $\tilde s$ can be glued from $s\in {\cal F}(U_1)$ and $0\in {\cal F}(U_2)$. }\endproof \definition Let ${\cal F}$ be a sheaf. Denote the space of sections of ${\cal F}$ on $U$ with compact support by ${\cal F}_c(U)$. Let ${\cal F}^*(U)$ map $U$ to the dual space ${\cal F}_c(U)^*$. Using the claim above, we obtain a restriction map ${\cal F}^*(V)\arrow {\cal F}^*(U)$ for each open $V\supset U$. This gives {\bf \blue dual presheaf} ${\cal F}^*$ \exercise Let $M$ be a manifold, and ${\cal F}$ a sheaf of modules over $C^\infty M$. {\bf \red Prove that ${\cal F}^*$ is a sheaf.} {\bf\green HINT:} Use partition of unity. \newpage {\bf \blue Rings and derivations (reminder)} \remark All rings in these handouts are assumed to be commutative and with unit. Algebras are associative, but not necessarily commutative (such as the matrix algebra). {\bf\blue Rings over a field $k$} are rings containing a field $k$. We assume that $k$ has characteristic 0. \definition Let $R$ be a ring over a field $k$. A $k$-linear map $D\; R \arrow R$ is called {\bf\blue a derivation} if it satisfies {\bf\blue the Leibnitz equation} $D(fg) = D(f) g + gD(f)$. The space of derivations is denoted as $\Der_k(R)$. \example $\frac{d}{dt}:\; \C[t]\arrow \C[t]$. \ \ $\frac{d}{dt}:\; C^\infty \R\arrow C^\infty\R$. \theorem Let $x_1, ..., x_n$ be coordinate functions on $\R^n$, $R=C^\infty\R^n$, and $\Der(R) \stackrel \Pi \arrow(C^\infty\R^n)^n$ map $D$ to $(D(x_1), D(x_2), ..., D(x_n))$. {\bf \red Then \[ \Pi:\; \Der(C^\infty\R^n) \arrow R^n\] is an isomorphism.} \newpage {\bf \blue Derivations of an algebra of compactly supported functions} {\bf \green SUBLEMMA:} Let $M$ be a smooth manifold, and $C^\infty_c(U)$ the space of compactly supported smooth functions, equipped with natural multiplication. Consider the space $\Der(M, C^\infty_c(M))$ of derivations of $C^\infty_c(M)$ with values in $C^\infty(M)$. Let $f$ be a function with support in $Z\subset M$. {\bf \red Then $D(f)$ has support in $Z$ for each $D\in Der(M, C^\infty_c(M))$.} {\bf \green Proof:} For each $g$ with support outside of $Z$, we have $0=D(fg) = f D(g) + g D(f)$, hence for each $U\subset M\backslash Z$, we have $0= g D(f)\restrict U$ whenever $\Sup(g)\subset U$. Then, $D(f)\restrict U=0$ as well. \endproof \lemma {\bf \red Let $\phi$ be a smooth function with compact support which is equal to 1 on $U\subset M$, and $D\in C^\infty_c(M)$. Then $D(\phi f)\restrict U= D(f)\restrict U$.} {\bf \green Proof. Step 1:} $D(\phi f)= D(\phi) f - \phi D(f)$, hence {\bf \purple it would suffice to prove that $D(\phi)\restrict U=0$.} {\bf \green Step 2:} Now, $\phi^2-\phi=0$ on $U$. By the Sublemma above, $0=D(\phi^2-\phi)\restrict U= [2D(\phi)\phi -D(\phi)]\restrict U= D(\phi)\restrict U$. \endproof \newpage {\bf \blue Derivations with compact support and without} \theorem Define the tautological map $\Der(C^\infty(M))\stackrel \tau \arrow \Der(M, C^\infty_c(M))$, taking $D$ to itself. {\bf \red Then $\tau$ is an isomorphism: \[ \Der(M, C^\infty_c(M))= \Der(C^\infty(M)).\]} \!\!\!{\bf \green Proof. Step 1:} For each $x\in M$, consider a smooth function $\psi_x$ with compact support which is equal to 1 in some neighbourhood $U_x\ni x$. Let $D\in \ker \tau$, and $f\in C^\infty M$. Then $0=D(\psi_x f)\restrict {U_x} = D(f)\restrict {U_x},$ hence $D(f)=0$ in a neighbourhood of any $x\in M$. {\bf \purple This implies that $\ker \tau =0$.} {\bf \green Step 2:} Consider $D\in \Der(M, C^\infty_c(M))$, and let $D(f)\restrict {U_x}:=D(\psi_x f)$. For any $U_x, U_y$, the Lemma above implies \[ D(\psi_x f)\restrict{U_x\cap U_y}= D(\psi_y f)\restrict{U_x\cap U_y} =D(f)\restrict{U_x\cap U_y} \] {\bf \purple Therefore, the sections $D(f)\restrict {U_x}$ agree on pairwise intersections, and define a section $\tilde D(f)\in C^\infty M$.} {\bf \green Step 3:} On germs, $\tilde D(f)=D(f)$, hence it satisfies Leibnitz rule on each germ, and therefore, {\bf \purple $\tilde D$ is a derivation} {\bf \red (exercise: check this directly).} {\bf \green Step 4:} By the Lemma above, $\tilde D(f)\restrict{U_x}=D(f)\restrict{U_x}$ for each $f$ with compact support. {\bf \purple Therefore, $\tau(\tilde D)=D$, and $\tau$ is surjective.} \endproof \newpage {\bf \blue Derivations as a sheaf} \definition Let $U\subset V$ be open subsets of a smooth manifold $M$, and $D\in \Der(V, C^\infty_c(V))$. For any $f\in C^\infty_c(U)$, extend $f$ to $\tilde f\in C^\infty_c(V)$ with the same support, (zero extension), and let $D\restrict U(f):= D(\tilde f)$. {\bf \blue This defines a structure of presheaf $U\arrow C^\infty_c(U)$.} \claim $U\arrow C^\infty_c(U)$ is a sheaf. {\bf\green Proof. Step 1:} A vector field is uniquely determined by its restriction to the germs of all sections, hence a derivation $D$ which vanishes on all germs for all $x\in M$ vanishes everywhere. {\bf \purple This takes care of the first sheaf axiom. } {\bf\green Proof. Step 2:} Let $\{U_i\}$ be a cover of $M$. To glue a derivation $D$ from its bits $D_i\in \Der(C^\infty(U_i))$, consider a partition of unity $\psi_i$ subordinate to $\{U_i\}$. {\bf \purple Then $D(f):= \sum D_i(\psi_i f)$ is a derivation which restricts to all $D_i$.} \endproof \corollary {\bf \purple The sheaf of derivations is locally free, that is, $\Der C^\infty M$ defines a vector bundle on $M$.} \definition It is called {\bf \blue the tangent bundle}, and denoted $TM$. \newpage {\bf \blue NEXT TIME.} {\bf \red \Huge Next time:\\[10mm] TEST ASSIGNMENT \# 2!} \end{document}