Are manifolds almost diffeomorphic to $mathbb{R}^n$?












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$begingroup$


Let $M$ be an $n$-dimensional compact connected orientable smooth manifold. Can we always find a diffeomorphism
$$Msetminus S_1congmathbb{R}^nsetminus S_2$$
where $S_1subset M$ and $S_2subsetmathbb{R}^n$ are subsets of measure zero?






general-topology differential-geometry manifolds smooth-manifolds







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  • $begingroup$
    Put any (smooth) Riemannian metric on $M$ and let $S_1=operatorname{Cut}(p)$...
    $endgroup$
    – user10354138
    Jan 12 at 13:29






  • 1




    $begingroup$
    This was discussed several times at MSE. math.stackexchange.com/questions/322027/…, math.stackexchange.com/questions/18083/…
    $endgroup$
    – Moishe Cohen
    Jan 12 at 14:07








  • 1




    $begingroup$
    Also math.stackexchange.com/questions/1838633/…
    $endgroup$
    – Moishe Cohen
    Jan 12 at 14:13
















3












$begingroup$


Let $M$ be an $n$-dimensional compact connected orientable smooth manifold. Can we always find a diffeomorphism
$$Msetminus S_1congmathbb{R}^nsetminus S_2$$
where $S_1subset M$ and $S_2subsetmathbb{R}^n$ are subsets of measure zero?






general-topology differential-geometry manifolds smooth-manifolds







share|cite|improve this question









$endgroup$












  • $begingroup$
    Put any (smooth) Riemannian metric on $M$ and let $S_1=operatorname{Cut}(p)$...
    $endgroup$
    – user10354138
    Jan 12 at 13:29






  • 1




    $begingroup$
    This was discussed several times at MSE. math.stackexchange.com/questions/322027/…, math.stackexchange.com/questions/18083/…
    $endgroup$
    – Moishe Cohen
    Jan 12 at 14:07








  • 1




    $begingroup$
    Also math.stackexchange.com/questions/1838633/…
    $endgroup$
    – Moishe Cohen
    Jan 12 at 14:13














3












3








3


0



$begingroup$


Let $M$ be an $n$-dimensional compact connected orientable smooth manifold. Can we always find a diffeomorphism
$$Msetminus S_1congmathbb{R}^nsetminus S_2$$
where $S_1subset M$ and $S_2subsetmathbb{R}^n$ are subsets of measure zero?






general-topology differential-geometry manifolds smooth-manifolds







share|cite|improve this question









$endgroup$




Let $M$ be an $n$-dimensional compact connected orientable smooth manifold. Can we always find a diffeomorphism
$$Msetminus S_1congmathbb{R}^nsetminus S_2$$
where $S_1subset M$ and $S_2subsetmathbb{R}^n$ are subsets of measure zero?






general-topology differential-geometry manifolds smooth-manifolds




general-topology differential-geometry manifolds smooth-manifolds






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 12 at 12:04









Simon ParkerSimon Parker

1,43021017




1,43021017












  • $begingroup$
    Put any (smooth) Riemannian metric on $M$ and let $S_1=operatorname{Cut}(p)$...
    $endgroup$
    – user10354138
    Jan 12 at 13:29






  • 1




    $begingroup$
    This was discussed several times at MSE. math.stackexchange.com/questions/322027/…, math.stackexchange.com/questions/18083/…
    $endgroup$
    – Moishe Cohen
    Jan 12 at 14:07








  • 1




    $begingroup$
    Also math.stackexchange.com/questions/1838633/…
    $endgroup$
    – Moishe Cohen
    Jan 12 at 14:13


















  • $begingroup$
    Put any (smooth) Riemannian metric on $M$ and let $S_1=operatorname{Cut}(p)$...
    $endgroup$
    – user10354138
    Jan 12 at 13:29






  • 1




    $begingroup$
    This was discussed several times at MSE. math.stackexchange.com/questions/322027/…, math.stackexchange.com/questions/18083/…
    $endgroup$
    – Moishe Cohen
    Jan 12 at 14:07








  • 1




    $begingroup$
    Also math.stackexchange.com/questions/1838633/…
    $endgroup$
    – Moishe Cohen
    Jan 12 at 14:13
















$begingroup$
Put any (smooth) Riemannian metric on $M$ and let $S_1=operatorname{Cut}(p)$...
$endgroup$
– user10354138
Jan 12 at 13:29




$begingroup$
Put any (smooth) Riemannian metric on $M$ and let $S_1=operatorname{Cut}(p)$...
$endgroup$
– user10354138
Jan 12 at 13:29




1




1




$begingroup$
This was discussed several times at MSE. math.stackexchange.com/questions/322027/…, math.stackexchange.com/questions/18083/…
$endgroup$
– Moishe Cohen
Jan 12 at 14:07






$begingroup$
This was discussed several times at MSE. math.stackexchange.com/questions/322027/…, math.stackexchange.com/questions/18083/…
$endgroup$
– Moishe Cohen
Jan 12 at 14:07






1




1




$begingroup$
Also math.stackexchange.com/questions/1838633/…
$endgroup$
– Moishe Cohen
Jan 12 at 14:13




$begingroup$
Also math.stackexchange.com/questions/1838633/…
$endgroup$
– Moishe Cohen
Jan 12 at 14:13










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