Bundles

## Bundles

### Grothendieck groups

Let $\mathrm{\pi }$ be an abelian category. The Grothendieck group of $\mathrm{\pi }$ is the group $K\left(\mathrm{\pi }\right)$ generated by

 $\left[M\right]\text{,}\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}M\beta \mathrm{\pi }\text{,}\phantom{\rule{2em}{0ex}}\text{with relations}\phantom{\rule{2em}{0ex}}\left[{M}_{1}\right]=\left[{M}_{2}\right]\text{,}\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}{M}_{1}\beta  {M}_{2}$
and
 $\left[M\right]=\left[{M}_{1}\right]+\left[{M}_{2}\right]\text{,}\phantom{\rule{1em}{0ex}}\text{if there exists an exact sequence}\phantom{\rule{1em}{0ex}}0\beta Ά{M}_{1}\beta ΆM\beta Ά{M}_{2}\beta Ά0.$
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 $E\stackrel{p}{\beta }X$ such that ${p}^{\beta 1}\left(x\right)\beta F$, for $x\beta X$, and if $x\beta X$ there is a neighborhood $U$ of $x$ and an isomorphism

$U Γ F βΆ Ο p β 1 ( U ) pr 1 β β p U = U$

Let $G$ be a group which acts on $F$. A $G$-bundle with fibre $F$ is a bundle $E\stackrel{p}{\beta }X$ such that the transition functions ${g}_{\mathrm{\Xi ±}\mathrm{\Xi ²}}:{U}_{\mathrm{\Xi ±}}\beta ©{U}_{\mathrm{\Xi ²}}\beta \mathrm{Aut}\left(F\right)$ given by

$g Ξ± Ξ² ( x ) = Ο Ξ± Ο Ξ² β 1 β£ { x } Γ F$

are morphisms ${g}_{\mathrm{\Xi ±}\mathrm{\Xi ²}}:{U}_{\mathrm{\Xi ±}}\beta ©{U}_{\mathrm{\Xi ²}}\beta G$. A principal $G$-bundle is a fibre bundle with fibre $G$ and a $G$-action $E\Gamma G\beta ΆE$.

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 $E\stackrel{p}{\beta }X$ such that ${p}^{\beta 1}\left(x\right)\beta V$, for $x\beta X$, and there is a open cover of $X$ and isomorphisms

$U Ξ± Γ V βΆ Ο Ξ± p β 1 ( U Ξ± ) pr 1 β β p U Ξ± = U Ξ±$

with $\mathrm{Ο}:x\Gamma V\beta {p}^{\beta 1}\left(x\right)$ a linear isomorphism. A vector bundle is a $\mathrm{GL}\left(V\right)$-bundle.

A section of $E$ is a morphism $s:X\beta E$

 $\begin{array}{c}E\\ p\beta \beta s\\ X\end{array}\phantom{\rule{2em}{0ex}}\text{such that}\phantom{\rule{2em}{0ex}}p\beta s={\mathrm{id}}_{X}$
The sheaf of sections of $E$ is given by with ${\mathrm{\pi ͺ}}_{X}\left(U\right)$-action given by $\left(fs\right)\left(x\right)=f\left(x\right)s\left(x\right),$ for $f\beta {\mathrm{\pi ͺ}}_{X}\left(U\right),s\beta \mathrm{\beta °}\left(U\right)$. 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

 $G\Gamma X\beta ΆX\phantom{\rule{2em}{0ex}}\text{such that}\phantom{\rule{2em}{0ex}}1x=x\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{g}_{1}\left({g}_{2}x\right)=\left({g}_{1}{g}_{2}\right)x,$
for $x\beta X$ and ${g}_{1},{g}_{2}\beta G$.

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

• If $x\beta X$ the fiber ${E}_{x}={p}^{\beta 1}\left(x\right)$ is isomorphic to ${\mathrm{\beta }}^{m}$,
• $E$ is locally trivial, i.e. If $x\beta X$ then there exists an open set $U$ containing $x$ and a homeomorphism  $\begin{array}{ccc}U\mathrm{\Gamma }F& \stackrel{\mathrm{Ο}}{\beta Ά}& {p}^{\beta 1}\left(U\right)\\ {\mathrm{pr}}_{1}\beta & & \beta p\\ U& =& U\end{array}$
such that if $u\beta U$ then $\mathrm{Ο}:{\mathrm{\beta }}^{m}\stackrel{\beta }{\beta }{p}^{\beta 1}\left(u\right)$,
• If $e\beta E$ and $g\beta G$ then $p\left(ge\right)=gp\left(e\right)$,
• If $g\beta G$ and $x\beta X$ then $g:{E}_{x}\beta {E}_{gx}$ is a morphism of vector spaces.

## Sheaves

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

such that if $\left\{{U}_{\mathrm{\Xi ±}}\right\}$ is an open cover of $U$ and ${f}_{\mathrm{\Xi ±}}\beta {\mathrm{\pi ͺ}}_{X}\left(U\right)$ are such that

such that ${f}_{\mathrm{\Xi ±}}=f{\beta £}_{{U}_{\mathrm{\Xi ±}}}$, for all $\mathrm{\Xi ±}$.

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

 ${A}^{\beta n}\beta M,\phantom{\rule{2em}{0ex}}\text{for some}\phantom{\rule{1em}{0ex}}n\beta {\mathrm{\beta €}}_{\beta ₯0}.$
An $A$-module $M$ is finitely generated, or of finite type, if there is a surjective morphism
 ${A}^{\beta n}\beta M,\phantom{\rule{2em}{0ex}}\text{for some}\phantom{\rule{1em}{0ex}}n\beta {\mathrm{\beta €}}_{\beta ₯0}.$
An $A$-module $M$ is finitely presented if there is an exact sequence

A locally free sheaf on $X$ is a sheaf $\mathrm{\beta ±}$ of ${\mathrm{\pi ͺ}}_{X}$-modules such that if $x\beta X$ then there is a neighborhood $U$ of $x$ with

 $\mathrm{\beta ±}\left(U\right)\beta {\mathrm{\pi ͺ}}_{X}^{\beta n}$.

A coherent sheaf on $X$ is a locally finitely presented sheaf of ${\mathrm{\pi ͺ}}_{X}$-modules, i.e. a sheaf $\mathrm{\beta ±}$ of ${\mathrm{\pi ͺ}}_{X}$-modules such that

• $\mathrm{\beta ±}$ is of finite type, i.e. $\mathrm{\beta ±}\left(U\right)$ is generated by a finite set of sections (there exists a surjection ${\mathrm{\pi ͺ}}_{X}\left(U{\right)}^{\beta n}\beta \mathrm{\beta ±}\left(U\right)$),
• For each open set $U$ of $X$ and each homomorphism ${\mathrm{\pi ͺ}}_{X}\left(U{\right)}^{\beta n}\stackrel{\mathrm{Ο}}{\beta }\mathrm{\beta ±}\left(U\right)$, $\mathrm{ker}\mathrm{Ο}$ is of finite type.

Let $G$ be a group that is also a space and let $X$ be a space with a $G$-action $G\Gamma X\stackrel{a}{\beta Ά}X.$ A $G$-equivariant sheaf is a sheaf $\mathrm{\beta ±}$ of ${\mathrm{\pi ͺ}}_{X}$-modules with an isomorphism

 ${a}^{*}\mathrm{\beta ±}\stackrel{\mathrm{Ο}}{\beta Ά}{\mathrm{pr}}_{2}^{*}\mathrm{\beta ±}\phantom{\rule{2em}{0ex}}\text{such that}\phantom{\rule{2em}{0ex}}{p}_{23}^{*}\mathrm{Ο}\beta \left({\mathrm{id}}_{G}\Gamma a{\right)}^{*}\mathrm{Ο}=\left(m\Gamma {\mathrm{id}}_{X}{\right)}^{*}\mathrm{Ο}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\mathrm{Ο}{\beta £}_{1\Gamma X}={\mathrm{id}}_{\mathrm{\beta ±}},$
where ${p}_{23}:G\Gamma G\Gamma X\beta ΆG\Gamma X$ is given by ${p}_{23}\left({g}_{1},{g}_{2},x\right)=\left({g}_{2},x\right)$ and we identify ${a}^{*}\mathrm{\beta ±}{\beta £}_{1\Gamma X}=\mathrm{\beta ±}$ and ${\mathrm{pr}}_{2}^{*}\mathrm{\beta ±}{\beta £}_{1\Gamma \mathrm{\beta ±}}=\mathrm{\beta ±}$.