The Lusternik–Schnirelmann bound on critical points #
A smooth function f on a compact boundaryless manifold M has at least as many critical points
as the unnormalized Lusternik–Schnirelmann category of M:
lsCategoryCount M univ ≤ totalCriticalCount I f
(TauCeti.lsCategoryCount_univ_le_totalCriticalCount). No nondegeneracy is assumed. Points are
counted, not critical values, and an infinite critical set has count ⊤. For nonempty M, whose
normalized category is cat M = lsCategoryCount M univ - 1, this says that f has at least
cat M + 1 critical points.
The bound is proved on the sublevels {f ≤ t}, by induction over the critical values
(TauCeti.lsCategoryCount_sublevel_le_encard). Across a compact band without critical points the
count of the sublevel does not grow (TauCeti.lsCategoryCount_sublevel_le_of_band): a deformation
of M along a descending flow carries the upper sublevel into the lower one. Across a critical
level c the deformation of TauCeti.exists_criticalLevelDeformation carries {f ≤ c + ε} into
{f < c - ε} ∪ U, for any open U containing the critical points of a band around c. Since a
manifold is locally contractible, U can be taken to be a union of one categorical open set per
critical point, so the count grows by at most the number of critical points in the band
(TauCeti.exists_lsCategoryCount_sublevel_le_add).
The bound also holds one compact open part at a time: the count of a compact open subset C, such
as a connected component of a compact manifold, is at most the number of critical points of f in
C (TauCeti.lsCategoryCount_univ_le_encard_criticalPoints_inter), since C is itself a compact
boundaryless manifold on which the critical points of the restriction of f are those of f. A
compact manifold has finitely many components. Their Lusternik–Schnirelmann counts add up to the
count of M (TauCeti.lsCategoryCount_univ_eq_finsum_connectedComponents) and their numbers of
critical points add up to that of f
(TauCeti.totalCriticalCount_eq_finsum_connectedComponents), so summing these bounds over the
components recovers the bound for M. In dimension zero the bound is an equality, both sides
being the number of points of M
(TauCeti.lsCategoryCount_univ_eq_totalCriticalCount_of_subsingleton).
Combined with the cup-length lower bound for the Lusternik–Schnirelmann category, on a nonempty
compact manifold every smooth function has at least lsCupLength M R + 1 critical points, for any
commutative ring R (TauCeti.lsCupLength_add_one_le_totalCriticalCount).
Main declarations #
TauCeti.lsCategoryCount_sublevel_le_of_band: the count of{f ≤ b}is at most that of{f ≤ a}whenf⁻¹[a, b]is compact and has no critical points.TauCeti.exists_lsCategoryCount_sublevel_le_add: the count of{f ≤ c + ε}exceeds that of{f ≤ c - ε}by at most the number of critical points in a compact band aroundc.TauCeti.lsCategoryCount_sublevel_le_encard: the count of{f ≤ t}is at most the number of critical points in{f ≤ t}.TauCeti.lsCategoryCount_univ_le_totalCriticalCount: the Lusternik–Schnirelmann theorem on a compact manifold.TauCeti.lsCategoryCount_univ_le_encard_criticalPoints_inter: the same bound for a compact open part ofM, such as a connected component of a compact manifold.TauCeti.lsCategoryCount_univ_eq_totalCriticalCount_of_subsingleton: equality in dimension zero.TauCeti.lsCupLength_add_one_le_totalCriticalCount: the cup-length bound on critical points.
References #
- L. Lusternik and L. Schnirelmann, Méthodes topologiques dans les problèmes variationnels, Hermann, Paris, 1934.
- O. Cornea, G. Lupton, J. Oprea and D. Tanré, Lusternik–Schnirelmann Category, Mathematical Surveys and Monographs 103, American Mathematical Society, 2003, Chapter 1.
The Lusternik–Schnirelmann count in dimension zero. On a manifold modelled on a
zero-dimensional space E every point is critical and every subset is open, so the
Lusternik–Schnirelmann count of M equals the number of critical points of any function f: both
are the number of points of M, possibly ⊤.
No growth across a band without critical points. If f is smooth and the band
f⁻¹[a, b] is compact and contains no critical point of f, then the Lusternik–Schnirelmann count
of the sublevel {f ≤ b} is at most that of {f ≤ a}.
Crossing a critical level. If f is smooth and the band f⁻¹[c - δ, c + δ], δ > 0, is
compact, then for some 0 < ε < δ the Lusternik–Schnirelmann count of {f ≤ c + ε} exceeds that
of {f ≤ c - ε} by at most the number of critical points of f in the band. The critical points
may be degenerate.
The Lusternik–Schnirelmann bound on sublevels. Let f be smooth and t < b, and suppose
that the sublevel {f ≤ b} is compact and contains only finitely many critical points of f. Then
the Lusternik–Schnirelmann count of {f ≤ t} is at most the number of critical points of f in
{f ≤ t}. The critical points may be degenerate.
The Lusternik–Schnirelmann theorem. On a compact boundaryless manifold M, the
Lusternik–Schnirelmann count of M is at most the number of critical points of any smooth
function on M. The critical points may be degenerate; if there are infinitely many, their count
is ⊤.
The Lusternik–Schnirelmann bound on a compact open part. If C is a compact open subset
of a boundaryless manifold M, for instance a connected component of a compact manifold, then the
Lusternik–Schnirelmann count of C, as a space in its own right, is at most the number of critical
points in C of a function f that is smooth on C. The critical points may be degenerate; if
there are infinitely many in C, their count is ⊤.
Summed over the finitely many components of a compact manifold M, using
TauCeti.lsCategoryCount_univ_eq_finsum_connectedComponents and
TauCeti.totalCriticalCount_eq_finsum_connectedComponents, these bounds recover
TauCeti.lsCategoryCount_univ_le_totalCriticalCount.
The cup-length bound on critical points. On a nonempty compact boundaryless manifold M,
every smooth function has at least lsCupLength M R + 1 critical points, for every commutative
ring R in the universe of M. The critical points may be degenerate; if there are infinitely
many, their count is ⊤.