definitions.tex

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\title{Useful theorems for Manifolds and Topology Preliminary Exams}
\author{University of Minnesota}

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\maketitle


\section{Manifolds}
	Unless otherwise stated, the following definitions come from \cite{lee-ism}.
	
	\begin{definition}
		A smooth map $F: M \rightarrow N$ is called a smooth immersion if its differential is injective at each point (equivalently if $\rk F= \dim M$)
	\end{definition}
	
	\begin{definition}
		Let $M$, $N$ be smooth manifolds with or without boundary. 
		A \textbf{smooth embedding of $M$ into $N$} is a smooth immersion $F:M \rightarrow N$ which is also a topological embedding, i.e. a homeomorphism onto its image $F(M)\subseteq N$ in the subspace topology.
		\textsc{Note:} A smooth embedding is both a topological embedding and a smooth immersion not just a topological embedding which happens to be smooth.
	\end{definition}
	


\section{Topology}
	Unless otherwise stated, the following definitions come from \cite{hatcher-at}.
	


\begin{thebibliography}{9}

\bibitem{hatcher-at}
Allen Hatcher.
\textit{Algebraic Topology}
Cambridge University Press, Cambridge, U.K., 2001

\bibitem{lee-ism} 
John M. Lee. 
\textit{Introduction to Smooth Manifolds}. 
Springer, New York, U.S.A., 2013

\end{thebibliography}


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