# nLab Sard's theorem

Named after Anthony P. Morse and Arthur Sard.

## Statement

Recall the following definitions from differential 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 $N$.

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.