regular covering
RegularCovering
2003-02-12 03:40:22
2004-01-24 13:34:41
Definition
%\documentclass{amsart}
\usepackage{amsmath}
%\usepackage[all,poly,knot,dvips]{xy}
%\usepackage{pstricks,pst-poly,pst-node,pstcol}
\usepackage{amssymb,latexsym}
\usepackage{amsthm,latexsym}
\usepackage{eucal,latexsym}
% THEOREM Environments --------------------------------------------------
\newtheorem{thm}{Theorem}
\newtheorem*{mainthm}{Main~Theorem}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{claim}[thm]{Claim}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\theoremstyle{remark}
\newtheorem{rem}[thm]{Remark}
\numberwithin{equation}{subsection}
%--------------------- Greek letters, etc -------------------------
\newcommand{\CA}{\mathcal{A}}
\newcommand{\CC}{\mathcal{C}}
\newcommand{\CM}{\mathcal{M}}
\newcommand{\CP}{\mathcal{P}}
\newcommand{\CS}{\mathcal{S}}
\newcommand{\BC}{\mathbb{C}}
\newcommand{\BN}{\mathbb{N}}
\newcommand{\BR}{\mathbb{R}}
\newcommand{\BZ}{\mathbb{Z}}
\newcommand{\FF}{\mathfrak{F}}
\newcommand{\FL}{\mathfrak{L}}
\newcommand{\FM}{\mathfrak{M}}
\newcommand{\Ga}{\alpha}
\newcommand{\Gb}{\beta}
\newcommand{\Gg}{\gamma}
\newcommand{\GG}{\Gamma}
\newcommand{\Gd}{\delta}
\newcommand{\GD}{\Delta}
\newcommand{\Ge}{\varepsilon}
\newcommand{\Gz}{\zeta}
\newcommand{\Gh}{\eta}
\newcommand{\Gq}{\theta}
\newcommand{\GQ}{\Theta}
\newcommand{\Gi}{\iota}
\newcommand{\Gk}{\kappa}
\newcommand{\Gl}{\lambda}
\newcommand{\GL}{\Lamda}
\newcommand{\Gm}{\mu}
\newcommand{\Gn}{\nu}
\newcommand{\Gx}{\xi}
\newcommand{\GX}{\Xi}
\newcommand{\Gp}{\pi}
\newcommand{\GP}{\Pi}
\newcommand{\Gr}{\rho}
\newcommand{\Gs}{\sigma}
\newcommand{\GS}{\Sigma}
\newcommand{\Gt}{\tau}
\newcommand{\Gu}{\upsilon}
\newcommand{\GU}{\Upsilon}
\newcommand{\Gf}{\varphi}
\newcommand{\GF}{\Phi}
\newcommand{\Gc}{\chi}
\newcommand{\Gy}{\psi}
\newcommand{\GY}{\Psi}
\newcommand{\Gw}{\omega}
\newcommand{\GW}{\Omega}
\newcommand{\Gee}{\epsilon}
\newcommand{\Gpp}{\varpi}
\newcommand{\Grr}{\varrho}
\newcommand{\Gff}{\phi}
\newcommand{\Gss}{\varsigma}
\newcommand{\Au}{ \operatorname{Aut}}
\def\co{\colon\thinspace}
\begin{thm}
Let $p\co E\to X$ be a covering map where $E$ and $X$ are connected and
locally path connected and let $X$ have a basepoint $*$. The following are
equivalent:
\begin{enumerate}
\item The action of $\Au(p)$, the group of covering transformations of
$p$, is transitive on the fiber $p^{-1}(*)$,
\item for some $e\in p^{-1}(*)$, $p_*\left(\pi_1(E,e)\right)$ is a
normal subgroup of $\pi_1(X,*)$, where $p_*$
denotes $\pi_1(p)$,
\item $\forall e,e'\in p^{-1}(*), \quad
p_*\left(\pi_1(E,e)\right)=p_* \left(\pi_1(E,e')\right)$,
\item there is a discrete group $G$ such that $p$ is a principal $G$-bundle.
\end{enumerate}
\end{thm}
All the elements for the proof of this theorem are contained in the articles
about the monodromy action and the deck transformations.
\begin{defn}
A covering with the properties described in the previous theorem is called
a \emph{regular} or \emph{normal} covering. The term \emph{Galois}
covering is also used sometimes.
\end{defn}