Remarks on smooth manifolds + v 0.17

Signed-off-by: Marcello Seri <[email protected]>

1-manifolds.tex

@@ -260,7 +260,8 @@ \section{Differentiable manifolds}
 \end{exercise}
 
 \begin{definition}\label{def:diffstr}
-  \marginnote{The union of all atlases in a differentiable structure is the \emph{unique} \emph{maximal} atlas in the equivalence class. There is a one-to-one correspondence between differentiable structures and maximal differentiable atlases: for convenience and to lighten the notation, from now on, we will always regard a differentiable structure as a differentiable maximal atlas without further comments.}
+  \marginnote{The union of all atlases in a differentiable structure is the \emph{unique} \emph{maximal} atlas in the equivalence class.
+  There is a one-to-one correspondence between differentiable structures and maximal differentiable atlases \cite[Proposition 1.17]{book:lee}: for convenience and to lighten the notation, from now on, we will always regard a differentiable structure as a differentiable maximal atlas without further comments.}
   A \emph{differentiable structure}, or more precisely a \emph{smooth structure}, on a topological manifold is an equivalence class of smooth atlases.
 \end{definition}
 
@@ -274,6 +275,8 @@ \section{Differentiable manifolds}
 \end{definition}
 
 In colloquial language, a differentiable manifold is just a space covered by charts with differentiable transition maps.
+Note that not all topological manifolds can be made into smooth manifolds, but counterexamples are hard to construct and you need at least to go to dimension 4.
+A nice and super compact explanation with the relevant reference is in \cite{SE2691140}.
 
 \begin{notation}
   Whenever possible we will omit the differentiable structure $\cA$ from the notation and just write $M$.

aom.bib

@@ -197,7 +197,15 @@ @misc{lectures:teufel
 @misc{nlab:urysohn_metrization_theorem,
   author       = {{nLab authors}},
   title        = {{{U}}rysohn metrization theorem},
-  url = {http://ncatlab.org/nlab/show/Urysohn%20metrization%20theorem/10},
+  url          = {http://ncatlab.org/nlab/show/Urysohn%20metrization%20theorem/10},
   month        = nov,
   year         = 2020
 }
+
+@MISC {SE2691140,
+  TITLE        = {Give me an example of a topological manifold which is not a smooth manifold.},
+  AUTHOR       = {C. Falcon},
+  HOWPUBLISHED = {Mathematics Stack Exchange},
+  NOTE         = {(version: 2018-03-14)},
+  URL          = {https://web.archive.org/web/20211117121257/https://math.stackexchange.com/questions/2691131/give-me-an-example-of-a-topological-manifold-which-is-not-a-smooth-manifold}
+}

aom.tex

@@ -6,9 +6,9 @@
 %Next block avoids bug, from  http://tex.stackexchange.com/a/200725/1913 
 \ifx\ifxetex\ifluatex\else % if lua- or xelatex http://tex.stackexchange.com/a/140164/1913
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@@ -213,7 +213,7 @@
   \setlength{\parskip}{\baselineskip}
   Copyright \copyright\ \the\year\ \thanklessauthor
 
-  \par Version 0.16 -- \today
+  \par Version 0.17 -- \today
 
   \vfill
   \small{\doclicenseThis}