The minimal number of critical points of a smooth function #
The invariant TauCeti.minCriticalCount I M of a manifold M is the least number of critical
points of a smooth real function on M, the number c(M) of Lusternik–Schnirelmann theory. It
takes values in ℕ∞ and minimizes over all smooth functions, with no Morse condition: a
function with infinitely many critical points contributes ⊤ and so never lowers the minimum.
Since ℕ∞ is well-ordered, the minimum is attained by a smooth function
(TauCeti.exists_totalCriticalCount_eq_minCriticalCount).
On a compact boundaryless manifold the minimum is finite (TauCeti.minCriticalCount_ne_top):
a Morse function exists, and it has finitely many critical points. Together with attainment this
makes c(M) a natural number, realized by a smooth function with a finite critical set; no default
natural value is assigned when the minimum is infinite. The minimum is bounded below by the
Lusternik–Schnirelmann count of M (TauCeti.lsCategoryCount_univ_le_minCriticalCount), hence,
when M is nonempty, by the cup length of M plus one
(TauCeti.lsCupLength_add_one_le_minCriticalCount); it
vanishes exactly on the empty manifold (TauCeti.minCriticalCount_eq_zero_iff), and it is at
least 2 as soon as M has two points (TauCeti.two_le_minCriticalCount), attained on every
sphere by the height function (TauCeti.minCriticalCount_sphere).
Main declarations #
TauCeti.minCriticalCount: the minimal number of critical points of a smooth function.TauCeti.minCriticalCount_leandTauCeti.le_minCriticalCount_iff: its characterization as the greatest lower bound of the critical counts of smooth functions.TauCeti.exists_totalCriticalCount_eq_minCriticalCount: the minimum is attained.TauCeti.minCriticalCount_ne_top: the minimum is finite on a compact manifold.TauCeti.lsCategoryCount_univ_le_minCriticalCount: the Lusternik–Schnirelmann lower bound.TauCeti.lsCupLength_add_one_le_minCriticalCount: the cup-length lower bound for nonemptyM.TauCeti.minCriticalCount_eq_zero_iffandTauCeti.two_le_minCriticalCount: the elementary lower bounds.TauCeti.minCriticalCount_sphere: the minimum is2on every sphere.
References #
- O. Cornea, G. Lupton, J. Oprea and D. Tanré, Lusternik–Schnirelmann Category, Mathematical Surveys and Monographs 103, American Mathematical Society, 2003, Chapter 1.
The minimal number of critical points c(M) of a smooth real function on M, in ℕ∞.
It is the infimum of TauCeti.totalCriticalCount I f over all smooth f, Morse or not; it is
attained (TauCeti.exists_totalCriticalCount_eq_minCriticalCount) and it is ⊤ exactly when
every smooth function has infinitely many critical points.
Instances For
Every smooth function has at least c(M) critical points.
A lower bound for the number of critical points of every smooth function is a lower bound
for c(M).
The minimal number of critical points is attained by a smooth function.
The empty manifold has c(M) = 0.
On a compact boundaryless manifold, c(M) = 0 exactly when M is empty: on a nonempty one
every continuous function has a critical point.
On a compact boundaryless manifold with at least two points, every smooth function has at
least two critical points, so 2 ≤ c(M).
On a compact boundaryless manifold the minimal number of critical points is finite: a Morse function exists, and it has finitely many critical points.
The Lusternik–Schnirelmann lower bound for c(M). On a compact boundaryless manifold,
every smooth function has at least as many critical points as the Lusternik–Schnirelmann count of
M, so c(M) is bounded below by that count.
The cup-length lower bound for c(M). On a nonempty compact boundaryless manifold M,
every smooth function has more critical points than the cup length of M with coefficients in any
commutative ring R in the universe of M, so lsCupLength M R + 1 ≤ c(M).
The minimal number of critical points of a sphere is 2. The height function has exactly
the two poles as critical points, and no smooth function on a compact manifold with at least two
points has fewer.