sheaf of sections SheafOfSections2 2006-03-17 13:12:05 2008-11-11 11:29:57 Definition vector bundle Sheaf of Sections sheaf vector bundle sections % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\spe}[1]{\ensuremath{ \text{Sp\'e}(#1)}} \newcommand{\RR}[0]{\ensuremath{\mathbb R}} \newcommand{\takes}[2]{\!:\!#1 \rightarrow #2} \subsection{Presheaf Definition} Consider a rank $r$ vector bundle $E\rightarrow M$, whose typical fibre is defined with respect to a field $k$. Let $\{U_\alpha\}$ constitute a cover for $M$. Then, sections of the bundle over some $U\subset M$ are defined as continuous functions $U\rightarrow E$, which commute with the natural projection map $\pi \takes E M$; $\pi \circ s = id_M$. Denote the space of sections of the bundle over U to be $\Gamma(U,E)$. The space of sections is a vector space over the field $k$ by defining addition and scalar multiplication pointwise: for $s,t \in \Gamma(U,E)$, $p\in U$ and $a\in k$ \[ (s+t)(p) \equiv s(p) + t(p) \qquad \qquad (a \cdot s) (p) \equiv a \cdot s(p).\] Then, this forms a presheaf $\mathcal E$, a functor from $((\text{top}_M))$ to the category of vector spaces, with restriction maps the natural restriction of functions. \subsection{Sheaf Axioms} It is easy to see that it satisfies the sheaf axioms: for $U$ open and $\{V_i\}$ a cover of $U$, \begin{enumerate} \item if $s\in \mathcal E(U)$ and $s\vert_{V_i} =0 $ for all $i$, then $s=0$. \item if $s_i\in \mathcal E(V_i)$ for all $i$, such that for each $i,j$ with $V_i\cap V_j \ne \emptyset$, $s_i\vert_{V_i\cap V_j} = s_j\vert_{V_i\cap V_j}$, then there is an $s \in \mathcal E(U)$ with $s\vert_{V_i} = s_i$ for all $i$. \end{enumerate} The first follows from the fact that for any $U$, there is always at least one element of $\mathcal E(U)$, the zero section, and that the transition functions of the bundle are linear maps. The second follows by the construction of the bundle.\\ \section{Sheafification} We may also see the vector bundle by applying associated sheaf construction to the presheaf $U \mapsto \Gamma(U,E)$. First though, we show that the stalk of the sheaf $\mathcal E$ at a point is isomorphic to the fibre of the bundle $E$ at the point. Let $[s,U]$ be a germ at $p\in M$ $(p\in U \subset M)$, and define a map $\psi \takes{\mathcal E_p}{ E_p}$ by \[\psi:[s,U] \mapsto s_p.\] First, we show that the map is a vector space homomorphism. Consider two germs $[s,U]$ and $[t,V]$ in $\mathcal E_p$. These map to $s_p$ and $t_p$ respectively. We add the germs by finding an open set $W\in U\cap V$ and adding the restrictions of the sections; \[[s,U] + [t,V] \equiv [s|_W + t|_W, W].\] Of course, $p\in W$, so we have $\psi(s|_W + t|_W) = s_p + t_p$, since the restriction maps are simply restriction of functions. Now, it is easy to show that $\psi$ is injective. Assume $\psi([t,V]) = \psi([s,U]) = s_p$. Then \begin{align*} \psi([t,V]) - \psi([s,U]) &= s_p - s_p \\ \psi([t,V] - [s,U]) &= 0\\ [t,V] &= [s,U] \end{align*} Now, we show that $\psi$ is surjective. For $s_p\in E_p$, let $U\subset M$ open be isomorphic to some subset $U_\RR$ of $\RR^m$. Then, $\Gamma(U,E)$ is the set of continuous maps $U\rightarrow V_E$, where $V_E$ is the typical fibre of $E$; \[\Gamma(U,E) = \bigoplus_{i=1}^r \mathcal C_{U_\RR}^\infty.\] Then let $[s,U]$ be the constant function $s:U_\RR\mapsto s_x$, and we have constructed an isomorphism $\psi$ between $\mathcal E_p$ and $E_p$.\\ To construct the \'Etal\'e space, take the disjoint union of stalks, \(\text{Sp\'e}(\mathcal E) = \coprod_{p\in M} \mathcal E_p\), and endow it with the following topology: the open sets shall be of the form \[ U_s = \bigl\{s_p | s\in \Gamma(U,\mathcal E), p\in U\subset M\bigr\},\] collection of germs of sections at points in $U\subset M$.\\ Then, the associated sheaf to $\mathcal E$ is the presheaf which assigns continuous maps $\Gamma(U,\spe {\mathcal E})$ to each open $U$. These are maps where the preimage of $U_s$ is open. Clearly, this implies that $\Gamma(U,E) \subset \Gamma(U,\spe{\mathcal E})$. To go the other way, note that open sets of $\spe{\mathcal E}$ are the images of continuous maps $U\rightarrow E$. An open subset of $\spe{\mathcal E}$ may be written as a union of $U_t$; $U_{ts} \equiv \{t_p, s_p | p\in U\}$. Then, by single-valuedness of maps, a continuous map $U\rightarrow \spe{\mathcal E}$ must map to $U_t$ for some $t\in \Gamma(U,E)$, so we have $\Gamma(U,E) \supset \Gamma(U,\spe{\mathcal E})$.