UniversalBundle 2002-11-01 17:23:37 2004-03-28 03:11:25 Definition classifying space \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{xypic} \newtheorem{thm}{Theorem} \newtheorem{prop}{Proposition} \newcommand{\ab}[1]{{#1}_{\mathrm{ab}}} \newcommand{\Ad}{\mathrm{Ad}} \newcommand{\ad}{\mathrm{ad}} \newcommand{\Aut}{\mathrm{Aut}\,} \newcommand{\Aff}[2]{\mathrm{Aff}_{#1} #2} \newcommand{\aff}[2]{\mathfrak{aff}_{#1} #2} \newcommand{\mcB}{\mathcal{B}} \newcommand{\bb}[1]{\mathbb{#1}} \newcommand{\bfrac}[2]{\left[\frac{#1}{#2}\\right]} \newcommand{\bkh}{\backslash} \newcommand{\Cyc}[2]{\mathcal{C}^{#1}_{#2}} \newcommand{\Cbar}[2]{\overline{\C{#1}{#2}}} %\newcommand{\CD}{\R[\Delta]} \newcommand{\C}{\mathbb{C}} \newcommand{\CF}[2]{\ensuremath{\mathfrak{C}(#1,#2)}} \newcommand{\Cinf}{\EuScript{C}^{\infty}} \newcommand{\cmp}{cyclic mod $p$\xspace} \newcommand{\cp}{\mathrm{c.p.}} \newcommand{\CS}{\EuScript{CS}} \newcommand{\deck}{\EuScript{D}} \newcommand{\defl}[1]{\mathfrak{def}_{#1}} \newcommand{\Der}{\mathrm{Der}\,} \newcommand{\eH}{[X_H]-[Y_H]} \newcommand{\EL}{\mathcal{EL}} \newcommand{\End}{\mathrm{End}} \newcommand{\ES}[1]{\EuScript{#1}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Fix}{\mathrm{Fix}} \newcommand{\fr}[1]{\mathfrak{#1}} \newcommand{\Frat}{\mathrm{Frat}\,} \newcommand{\Gal}[1]{\Gamma(#1 |\Q)} \newcommand{\GL}[2]{\mathrm{GL}_{#1} #2} \newcommand{\gl}[2]{\mathfrak{gl}_{#1} #2} \newcommand{\GrR}[1]{a(#1 G)} \newcommand{\Gr}{\mathrm{Gr}\,} \newcommand{\mcH}{\mathcal{H}} \renewcommand{\H}{\mathbb{H}} \newcommand{\Hom}[2]{\mathrm{Hom}(#1,#2)} \newcommand{\id}{\mathrm{id}} \newcommand{\im}{\mathrm{im}} \newcommand{\ind}[2]{\mathrm{ind}^{#1}_{#2}} \newcommand{\indp}[2]{\mathfrak{ind}^{#1}_{#2}} \renewcommand{\inf}[1]{\mathfrak{inf}_{#1}} \newcommand{\inn}[1]{\langle #1\rangle} \renewcommand{\int}{\mathrm{int}} \newcommand{\Iso}{\mathrm{Iso}} \newcommand{\K}{\mathcal{K}} \renewcommand{\ker}{\mathrm{ker}\,} \renewcommand{\L}[1]{\mathfrak{L}(#1)} \newcommand{\lap}[1]{\Delta_{#1}} \newcommand{\lapM}{\Delta_M} \newcommand{\Lie}{\mathrm{Lie}} \newcommand{\lineq}{linearly equivalent\xspace} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mG}{m_G} \newcommand{\mK}{m_{\K}} \newcommand{\mindeg}[1]{\fr{md}(#1)} \newcommand{\N}{\mathbb{N}} \renewcommand{\O}{\mathcal{O}} \newcommand{\Om}{\Omega} \newcommand{\om}{\omega} \newcommand{\Orb}{\mathrm{Orb}} \newcommand{\pad}{\hat{\Z}_p} \newcommand{\pder}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\pderw}[1]{\frac{\partial}{\partial #1}} \newcommand{\pdersec}[2]{\frac{\partial^2 #1}{\partial {#2}^2}} \newcommand{\perm}[1]{\pi_{#1}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\rad}{\mathrm{rad}\,} \newcommand{\res}[2]{\mathrm{res}^{#1}_{#2}} \newcommand{\resp}[2]{\mathfrak{res}^{#1}_{#2}} \newcommand{\RG}{\EuScript{R}_G} \newcommand{\rk}{\mathrm{rk}\,} \newcommand{\V}[1]{\mathbf{#1}} \newcommand{\vp}{\varphi} \newcommand{\Stab}{\mathrm{Stab}} \newcommand{\SL}[2]{\mathrm{SL}_{#1} #2} \renewcommand{\sl}[2]{\fr{sl}_{#1} #2} \newcommand{\SO}[2]{\mathrm{SO}_{#1} #2} \newcommand{\so}[2]{\fr{so}_{#1} #2} \newcommand{\Sp}[2]{\mathrm{Sp}_{#1} #2} \renewcommand{\sp}[2]{\fr{sp}_{#1} #2} \newcommand{\SU}[1]{\mathrm{SU}( #1)} \newcommand{\su}[1]{\fr{su}_{#1}} \newcommand{\Sym}{\mathrm{Sym}} \newcommand{\sym}{\mathrm{sym}} \newcommand{\Tg}{\mc{T}(\fr g)} \newcommand{\tom}{\tilde{\omega}} \newcommand{\ghtghp}{\fr g/\fr h\oplus(\fr g/\fr h^\perp)^*} \newcommand{\ghps}{(\fr g/\fr h^\perp)^*} \newcommand{\Tr}{\mathrm{Tr}} \newcommand{\tr}{\mathrm{tr}} %\renewcommand{\thechapter}{\Roman{chapter}} %\renewcommand{\thesection}{\thechapter.\arabic{section}} %\renewcommand{\thethm}{\thechapter.\arabic{thm}} \newcommand{\Ug}{\mc{U}(\fr g)} \newcommand{\Uh}{\mc{U}(\fr h)} \renewcommand{\V}[1]{\mathbf{#1}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Zp}{\Z/p} Let $G$ be a topological group. A \emph{universal bundle} for $G$ is a principal bundle $p :EG \to BG$ such that for any principal bundle $\pi:E\to B$, with $B$ a CW-complex, there is a map $\vp:B\to BG$, unique up to homotopy, such that the pullback bundle $\vp^*(p)$ is equivalent to $\pi$, that is such that there is a bundle map $\vp'$. $$\xymatrix{E\ar[d]^{\pi}\ar[r]^{\vp'(E)}&EG\ar[d]^p\\ B\ar[r]^{\vp'(B)}&BG}$$ with $\vp'(B)=\vp$, such that any bundle map of any bundle over $B$ extending $\vp$ factors uniquely through $\vp'$. As is obvious from the universal property, the universal bundle for a group $G$ is unique up to unique homotopy equivalence. The base space $BG$ is often called a {\em classifying space} of $G$, since homotopy classes of maps to it from a given space classify $G$-bundles over that space. There is a useful criterion for universality: a bundle is universal if and only if all the homotopy groups of $EG$, its total space, are trivial. This allows us to construct the universal bundle any subgroup from that of a larger group. Assume $H\leq G$ and that $p:EG\to BG$ is a universal bundle for $G$. Then $H$ also acts freely on $EG$ which is contractable so $p_H:EH=EB\to BH=EB/H$ must be a universal bundle for $H$. In 1956, John Milnor gave a general construction of the universal bundle for any topological group $G$ (see \emph{Annals of Mathematics}, Second Series, Volume 63 Issue 2 and Issue 3 for details). His construction uses the infinite join of the group $G$ with itself to define the total space of the universal bundle.