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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 17th 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: Named after [[Anthony P. Morse]] and [[Arthur Sard]]. ## Statement Recall the following definitions from [[diferential topology]]. The set of [[critical points]] of a smooth map $f$ is the set of points in the domain of $f$ where the tangent map is not surjective. The set of [[critical values]] of $f$ is the $f$-image of the set of [[critical points]] of $f$. The set of [[regular values]] of $f$ is the complement of the set of [[critical values]] of $f$. Suppose $M$ and $N$ are [[smooth manifolds]] of dimension $m$ and $n$ respectively and $f\colon M\to N$ is a $\mathrm{C}^r$-[[smooth map]], where $r\ge1$ and $r\gt m-n$. Then the set of critical values in $N$ is a [[meager subset]] (alias first category subset) and a [[negligible subset]] (alias measure zero subset) of $N$. In particular, the set of [[regular values]] is [[dense]] in $N$. Furthermore, the $f$-image of points of $M$ where $f$ has rank at most $r$ ($0\lt r\lt m$) has [[Hausdorff dimension]] at most $r$. If $N$ is a [[Banach manifold]] and $q\ge1$, $f$ is a Fredholm map, and $q$ is strictly greater than the index of $f$, then the critical values of $f$ form a [[meager subset]] of $N$. ## Related concepts * [[Thom's transversality theorem]] ## References The case $n=1$: * [[Anthony P. Morse]], _The Behavior of a Function on Its Critical Set_, Annals of Mathematics 40:1 (1939), 62–70. [doi](http://dx.doi.org/10.2307/1968544). The case $n\gt1$: * [[Arthur Sard]], _The measure of the critical values of differentiable maps_, Bulletin of the American Mathematical Society, 48:12 (1942), 883–890, [doi](https://doi.org/10.1090/S0002-9904-1942-07811-6). The case when $N$ is a [[Banach manifold]]: * [[Stephen Smale]], _An Infinite Dimensional Version of Sard's Theorem_, American Journal of Mathematics 87:4 (1965), 861–866. [doi](http://dx.doi.org/10.2307/2373250). The part concerning [[Hausdorff measures]]: * [[Arthur Sard]], _Hausdorff Measure of Critical Images on Banach Manifolds_, American Journal of Mathematics 87:1 (1965), 158–174. [doi](http://dx.doi.org/10.2307/2373229). <a href="https://ncatlab.org/nlab/revision/diff/Sard%27s+theorem/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Sard%27s+theorem/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Sard%27s+theorem">current</a>

   Added:

   Named after Anthony P. Morse and Arthur Sard.

   Statement

   Recall the following definitions from diferential topology. The set of critical points of a smooth map $f$ is the set of points in the domain of $f$ where the tangent map is not surjective. The set of critical values of $f$ is the $f$-image of the set of critical points of $f$. The set of regular values of $f$ is the complement of the set of critical values of $f$.

   Suppose $M$ and $N$ are smooth manifolds of dimension $m$ and $n$ respectively and $f\colon M\to N$ is a $\mathrm{C}^r$-smooth map, where $r\ge1$ and $r\gt m-n$. Then the set of critical values in $N$ is a meager subset (alias first category subset) and a negligible subset (alias measure zero subset) of $N$. In particular, the set of regular values is dense in $N$. Furthermore, the $f$-image of points of $M$ where $f$ has rank at most $r$ ($0\lt r\lt m$) has Hausdorff dimension at most $r$.

   If $N$ is a Banach manifold and $q\ge1$, $f$ is a Fredholm map, and $q$ is strictly greater than the index of $f$, then the critical values of $f$ form a meager subset of $N$.

   Related concepts

   • Thom’s transversality theorem

   References

   The case $n=1$:

   • Anthony P. Morse, The Behavior of a Function on Its Critical Set, Annals of Mathematics 40:1 (1939), 62–70. doi.

   The case $n\gt1$:

   • Arthur Sard, The measure of the critical values of differentiable maps, Bulletin of the American Mathematical Society, 48:12 (1942), 883–890, doi.

   The case when $N$ is a Banach manifold:

   • Stephen Smale, An Infinite Dimensional Version of Sard’s Theorem, American Journal of Mathematics 87:4 (1965), 861–866. doi.

   The part concerning Hausdorff measures:

   • Arthur Sard, Hausdorff Measure of Critical Images on Banach Manifolds, American Journal of Mathematics 87:1 (1965), 158–174. doi.

   diff, v3, current