Bundles

## Bundles

### Grothendieck groups

Let $𝒜$ be an abelian category. The Grothendieck group of $𝒜$ is the group $K\left(𝒜\right)$ generated by

 $\left[M\right]\text{,}\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}M\in 𝒜\text{,}\phantom{\rule{2em}{0ex}}\text{with relations}\phantom{\rule{2em}{0ex}}\left[{M}_{1}\right]=\left[{M}_{2}\right]\text{,}\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}{M}_{1}\cong {M}_{2}$
and
 $\left[M\right]=\left[{M}_{1}\right]+\left[{M}_{2}\right]\text{,}\phantom{\rule{1em}{0ex}}\text{if there exists an exact sequence}\phantom{\rule{1em}{0ex}}0⟶{M}_{1}⟶M⟶{M}_{2}⟶0.$
Let $X$ be a space and let $\mathrm{pt}$ be the space with a single point.
• The representation ring, or character ring, of $G$ is the Grothendieck group ${K}_{G}\left(\mathrm{pt}\right)$ of the category of $G$-modules.
• The (topological) $K$-theory of $X$ is the Grothendieck group $K\left(X\right)$ of the category of vector bundles on $X$.
• The (algebraic) $K$-theory of $X$ is the Grothendieck group $K\left(X\right)$ of the category of coherent sheaves on $X$.
• The (topological) $G$-equivariant $K$-theory of $X$ is the Grothendieck group ${K}_{G}\left(X\right)$ of the category of $G$-equivariant vector bundles on $X$.
• The (algebraic) $G$-equivariant $K$-theory of $X$ is the Grothendieck group ${K}_{G}\left(X\right)$ of the category of $G$-equivariant coherent sheaves on $X$.

### Bundles

Let $X$ and $F$ be spaces. A fibre bundle on $X$ with fiber $F$ is a space $E$ with a surjective map $E\stackrel{p}{\to }X$ such that ${p}^{–1}\left(x\right)\simeq F$, for $x\in X$, and if $x\in X$ there is a neighborhood $U$ of $x$ and an isomorphism

$U × F ⟶ φ p – 1 ( U ) pr 1 ↓ ↓ p U = U$

Let $G$ be a group which acts on $F$. A $G$-bundle with fibre $F$ is a bundle $E\stackrel{p}{\to }X$ such that the transition functions ${g}_{\alpha \beta }:{U}_{\alpha }\cap {U}_{\beta }\to \mathrm{Aut}\left(F\right)$ given by

$g α β ( x ) = φ α φ β – 1 ∣ { x } × F$

are morphisms ${g}_{\alpha \beta }:{U}_{\alpha }\cap {U}_{\beta }\to G$. A principal $G$-bundle is a fibre bundle with fibre $G$ and a $G$-action $E×G⟶E$.

Let $X$ be a space and let $V$ be a vector space. A vector bundle on $X$ with fiber $V$ is a space $E$ with a surjective map $E\stackrel{p}{\to }X$ such that ${p}^{–1}\left(x\right)\simeq V$, for $x\in X$, and there is a open cover of $X$ and isomorphisms

$U α × V ⟶ φ α p – 1 ( U α ) pr 1 ↓ ↓ p U α = U α$

with $\phi :x×V\to {p}^{–1}\left(x\right)$ a linear isomorphism. A vector bundle is a $\mathrm{GL}\left(V\right)$-bundle.

A section of $E$ is a morphism $s:X\to E$

 $\begin{array}{c}E\\ p↓↑s\\ X\end{array}\phantom{\rule{2em}{0ex}}\text{such that}\phantom{\rule{2em}{0ex}}p\circ s={\mathrm{id}}_{X}$
The sheaf of sections of $E$ is given by with ${𝒪}_{X}\left(U\right)$-action given by $\left(fs\right)\left(x\right)=f\left(x\right)s\left(x\right),$ for $f\in {𝒪}_{X}\left(U\right),s\in ℰ\left(U\right)$. Then is an equivalence of categories.

Let $G$ be a topological group. A $G$-space is a topological space $X$ with a $G$-action. More precisely, a $G$-space is a topological space $X$ with a continuous map

 $G×X⟶X\phantom{\rule{2em}{0ex}}\text{such that}\phantom{\rule{2em}{0ex}}1x=x\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{g}_{1}\left({g}_{2}x\right)=\left({g}_{1}{g}_{2}\right)x,$
for $x\in X$ and ${g}_{1},{g}_{2}\in G$.

Let $X$ be a $G$-space. A $G$-equivariant vector bundle on $X$ is a $G$-space $E$ with a surjective map $E\stackrel{p}{\to }X$ such that

• If $x\in X$ the fiber ${E}_{x}={p}^{–1}\left(x\right)$ is isomorphic to ${ℂ}^{m}$,
• $E$ is locally trivial, i.e. If $x\in X$ then there exists an open set $U$ containing $x$ and a homeomorphism  $\begin{array}{ccc}U×F& \stackrel{\phi }{⟶}& {p}^{–1}\left(U\right)\\ {\mathrm{pr}}_{1}↓& & ↓p\\ U& =& U\end{array}$
such that if $u\in U$ then $\phi :{ℂ}^{m}\stackrel{\simeq }{\to }{p}^{–1}\left(u\right)$,
• If $e\in E$ and $g\in G$ then $p\left(ge\right)=gp\left(e\right)$,
• If $g\in G$ and $x\in X$ then $g:{E}_{x}\to {E}_{gx}$ is a morphism of vector spaces.

## Sheaves

Let $X$ be a topological space. A sheaf on $X$ is a contravariant functor

such that if $\left\{{U}_{\alpha }\right\}$ is an open cover of $U$ and ${f}_{\alpha }\in {𝒪}_{X}\left(U\right)$ are such that

such that ${f}_{\alpha }=f{\mid }_{{U}_{\alpha }}$, for all $\alpha$.

Let $A$ be a ring. An $A$-module is free if there is an isomorphism

 ${A}^{\oplus n}\simeq M,\phantom{\rule{2em}{0ex}}\text{for some}\phantom{\rule{1em}{0ex}}n\in {ℤ}_{\ge 0}.$
An $A$-module $M$ is finitely generated, or of finite type, if there is a surjective morphism
 ${A}^{\oplus n}\to M,\phantom{\rule{2em}{0ex}}\text{for some}\phantom{\rule{1em}{0ex}}n\in {ℤ}_{\ge 0}.$
An $A$-module $M$ is finitely presented if there is an exact sequence

A locally free sheaf on $X$ is a sheaf $ℱ$ of ${𝒪}_{X}$-modules such that if $x\in X$ then there is a neighborhood $U$ of $x$ with

 $ℱ\left(U\right)\simeq {𝒪}_{X}^{\oplus n}$.

A coherent sheaf on $X$ is a locally finitely presented sheaf of ${𝒪}_{X}$-modules, i.e. a sheaf $ℱ$ of ${𝒪}_{X}$-modules such that

• $ℱ$ is of finite type, i.e. $ℱ\left(U\right)$ is generated by a finite set of sections (there exists a surjection ${𝒪}_{X}\left(U{\right)}^{\oplus n}\to ℱ\left(U\right)$),
• For each open set $U$ of $X$ and each homomorphism ${𝒪}_{X}\left(U{\right)}^{\oplus n}\stackrel{\phi }{\to }ℱ\left(U\right)$, $\mathrm{ker}\phi$ is of finite type.

Let $G$ be a group that is also a space and let $X$ be a space with a $G$-action $G×X\stackrel{a}{⟶}X.$ A $G$-equivariant sheaf is a sheaf $ℱ$ of ${𝒪}_{X}$-modules with an isomorphism

 ${a}^{*}ℱ\stackrel{\phi }{⟶}{\mathrm{pr}}_{2}^{*}ℱ\phantom{\rule{2em}{0ex}}\text{such that}\phantom{\rule{2em}{0ex}}{p}_{23}^{*}\phi \circ \left({\mathrm{id}}_{G}×a{\right)}^{*}\phi =\left(m×{\mathrm{id}}_{X}{\right)}^{*}\phi \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\phi {\mid }_{1×X}={\mathrm{id}}_{ℱ},$
where ${p}_{23}:G×G×X⟶G×X$ is given by ${p}_{23}\left({g}_{1},{g}_{2},x\right)=\left({g}_{2},x\right)$ and we identify ${a}^{*}ℱ{\mid }_{1×X}=ℱ$ and ${\mathrm{pr}}_{2}^{*}ℱ{\mid }_{1×ℱ}=ℱ$.