sheaf of sections

0.1 Presheaf Definition

Consider a rank $r$ vector bundle $E\to M$, whose typical fibre is defined with respect to a field $k$. Let $\{U_{\alpha }\}$ constitute a coverfor $M$. Then, sections of the bundle over some $U\subset M$ are definedas continuous functions $U\to E$, which commute with the natural projectionmap $\pi :E\to M$; $\pi \circ s=i{d}_M$. Denote the space of sections ofthe bundle over U to be $\mathrm{\Gamma }(U,E)$. The space of sections is a vectorspace over the field $k$ by defining addition and scalar multiplication pointwise: for $s,t\in \mathrm{\Gamma }(U,E)$, $p\in U$ and $a\in k$

$$(s+t)(p)\equiv s(p)+t(p)\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}(a\cdot s)(p)\equiv a\cdot s(p).$$

Then, this forms a presheaf $ℰ$, a functor from $(({\text{top}}_M))$to the category of vector spaces, with restriction maps the naturalrestriction of functions.

0.2 Sheaf Axioms

It is easy to see that it satisfies the sheafaxioms: for $U$ open and $\{V_i\}$ a cover of $U$,

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

2. if $s_i\in ℰ(V_i)$ for all $i$, such that for each $i,j$ with$V_i\cap V_j\ne \mathrm{\varnothing }$, ${s_i|}_{V_i\cap V_j}={s_j|}_{V_i\cap V_j}$, then there is an $s\in ℰ(U)$ with ${s|}_{V_i}=s_i$ forall $i$.

The first follows from the fact that for any $U$, there is always at leastone element of $ℰ(U)$, the zero section, and that the transitionfunctions 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 constructionto the presheaf $U\mapsto \mathrm{\Gamma }(U,E)$. First though, we show that thestalk of the sheaf $ℰ$ at a point is isomorphic to the fibre of thebundle $E$ at the point. Let $[s,U]$ be a germ at $p\in M$ $(p\in U\subset M)$, and define a map $\psi :{ℰ}_p\to E_p$ by

$$\psi :[s,U]\mapsto s_p.$$

First, we show that the map is a vector space homomorphism. Consider twogerms $[s,U]$ and $[t,V]$ in ${ℰ}_p$. These map to $s_p$ and $t_p$respectively. We add the germs by finding an open set $W\in U\cap V$ andadding 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 therestriction 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

$\psi ([t,V])-\psi ([s,U])$$=s_p-s_p$

$\psi ([t,V]-[s,U])$$=0$

$=[s,U]$

Now, we show that $\psi$ is surjective. For $s_p\in E_p$, let $U\subset M$open be isomorphic to some subset $U_{ℝ}$ of ${ℝ}^m$. Then, $\mathrm{\Gamma }(U,E)$ is theset of continuous maps $U\to V_E$, where $V_E$ is the typical fibre of $E$;

$$\mathrm{\Gamma }(U,E)=\underset{i=1}{\overset{r}{\oplus }}{𝒞}_{U_{ℝ}}^{\mathrm{\infty }}.$$

Then let $[s,U]$ be the constant function $s:U_{ℝ}\mapsto s_x$, and wehave constructed an isomorphism $\psi$ between ${ℰ}_p$ and $E_p$.

To construct the Étalé space, take the disjoint union of stalks,$\text{Spé}(ℰ)={\coprod }_{p\in M}{ℰ}_p$, and endow it withthe following topology: the open sets shall be of the form

$$U_s=\{s_p|s\in \mathrm{\Gamma }(U,ℰ),p\in U\subset M\},$$

collection of germs of sections at points in $U\subset M$.

Then, the associated sheaf to $ℰ$ is the presheaf which assignscontinuous maps $\mathrm{\Gamma }(U,\text{Spé}(ℰ))$ to each open $U$. These aremaps where the preimage of $U_s$ is open. Clearly, this implies that$\mathrm{\Gamma }(U,E)\subset \mathrm{\Gamma }(U,\text{Spé}(ℰ))$. To go the other way, notethat open sets of $\text{Spé}(ℰ)$ are the images of continuous maps $U\to E$. An open subset of $\text{Spé}(ℰ)$ 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 $U\to \text{Spé}(ℰ)$ must map to $U_t$ for some $t\in \mathrm{\Gamma }(U,E)$, so we have $\mathrm{\Gamma }(U,E)\supset \mathrm{\Gamma }(U,\text{Spé}(ℰ))$.