From 4c2ba41c2b706908dde39e637f194bbb0bdac7ac Mon Sep 17 00:00:00 2001 From: Marcello Seri Date: Wed, 5 Jan 2022 23:30:42 +0100 Subject: [PATCH] Update Signed-off-by: Marcello Seri --- 6-differentiaforms.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/6-differentiaforms.tex b/6-differentiaforms.tex index a702dd0..4090d9f 100644 --- a/6-differentiaforms.tex +++ b/6-differentiaforms.tex @@ -916,7 +916,7 @@ \section{De Rham cohomology and Poincar\'e lemma} In fact this is the case, thanks to the following theorem\footnote{This is a deep result related to the Whitney Embedding Theorem from Remark~\ref{rmk:WhitneyET} and is out of the scope of our course, for more details refer to~\cite[Chapter 6 and Theorems 6.26 and 9.27]{book:lee}.}. % \begin{theorem}[Whitney Approximation Theorem for continuous maps]\label{thm:WhitneyApproxCont} -Given any continuous mapping $G \in C^0(M,N)$, there exists $F \in C^\infty(M,N)$ which is homotopic to $G$. Moreover, if $G$ is smooth\footnote{Note that a function $f : M \to N$ is defined to be smooth on a subset $A \subset M$ if there is some smooth function $g: U \to N$, defined on an open $U\supset A$ such that $g = f$ on $A$.} on a closed subset $U\subset M$, then one can choose $F$ so that $F=G$ on $U$. +Given any continuous mapping $G \in C^0(M,N)$, there exists $F \in C^\infty(M,N)$ which is homotopic to $G$. Moreover, if $G$ is smooth\footnote{Note that a function $f : M \to N$ is defined to be smooth on a subset $A \subset M$ if there is some smooth function $g: U \to N$, defined on an open $U\supset A$ such that $g = f$ on $A$.} on a closed subset $A\subset M$, then one can choose $F$ so that $F=G$ on $A$. \end{theorem} % In particular, if two smooth maps are homotopic then they are also smoothly homotopic: we can assume the map $K$ to be smooth.