# sheaf of sections

### 0.1 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 $\pi\!:\!E\rightarrow M$; $\pi\circ s=id_{M}$. 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.

### 0.2 Sheaf Axioms

It is easy to see that it satisfies the sheaf axioms: for $U$ open and $\{V_{i}\}$ a cover of $U$,

1. 1.

if $s\in\mathcal{E}(U)$ and $s|_{V_{i}}=0$ for all $i$, then $s=0$.

2. 2.

if $s_{i}\in\mathcal{E}(V_{i})$ for all $i$, such that for each $i,j$ with $V_{i}\cap V_{j}\neq\emptyset$, $s_{i}|_{V_{i}\cap V_{j}}=s_{j}|_{V_{i}\cap V_{j}}$, then there is an $s\in\mathcal{E}(U)$ with $s|_{V_{i}}=s_{i}$ for all $i$.

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.

## 1 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 $\psi\!:\!\mathcal{E}_{p}\rightarrow E_{p}$ 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

 $\displaystyle\psi([t,V])-\psi([s,U])$ $\displaystyle=s_{p}-s_{p}$ $\displaystyle\psi([t,V]-[s,U])$ $\displaystyle=0$ $\displaystyle=[s,U]$

Now, we show that $\psi$ is surjective. For $s_{p}\in E_{p}$, let $U\subset M$ open be isomorphic to some subset $U_{\mathbb{R}}$ of $\mathbb{R}^{m}$. Then, $\Gamma(U,E)$ is the set of continuous maps $U\rightarrow V_{E}$, where $V_{E}$ is the typical fibre of $E$;

 $\Gamma(U,E)=\bigoplus_{i=1}^{r}\mathcal{C}_{U_{\mathbb{R}}}^{\infty}.$

Then let $[s,U]$ be the constant function $s:U_{\mathbb{R}}\mapsto s_{x}$, and we have constructed an isomorphism $\psi$ between $\mathcal{E}_{p}$ and $E_{p}$.

To construct the Étalé 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 $\Gamma(U,\text{Sp\'{e}}(\mathcal{E}))$ to each open $U$. These are maps where the preimage of $U_{s}$ is open. Clearly, this implies that $\Gamma(U,E)\subset\Gamma(U,\text{Sp\'{e}}(\mathcal{E}))$. To go the other way, note that open sets of $\text{Sp\'{e}}(\mathcal{E})$ are the images of continuous maps $U\rightarrow E$. An open subset of $\text{Sp\'{e}}(\mathcal{E})$ 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 $U\rightarrow\text{Sp\'{e}}(\mathcal{E})$ must map to $U_{t}$ for some $t\in\Gamma(U,E)$, so we have $\Gamma(U,E)\supset\Gamma(U,\text{Sp\'{e}}(\mathcal{E}))$.

Title sheaf of sections SheafOfSections 2013-03-22 15:46:36 2013-03-22 15:46:36 guffin (12505) guffin (12505) 7 guffin (12505) Definition msc 55R25 VectorBundle Sheaf of Sections