Deformation of sublevels #
Let f be a smooth real function on a boundaryless finite-dimensional manifold M, and let
a ≤ b be such that the band f⁻¹[a, b] is compact and contains no critical point of f. Then
the sublevel M_b = {f ≤ b} deformation retracts onto M_a = {f ≤ a}, by a homotopy fixing
M_a pointwise (TauCeti.exists_sublevelDeformation); in particular the inclusion of M_a into
M_b is a homotopy equivalence (TauCeti.exists_sublevelHomotopyEquiv). The manifold itself need
not be compact, nor σ-compact.
Since M_a is a strong deformation retract of M_b, any two retractions of M_b onto M_a are
homotopic relative to M_a (TauCeti.homotopicRel_sublevel_retraction). So the retraction is
determined up to such homotopy by f, a and b alone, whatever vector field or metric produced
it. In particular, for a ≤ a' ≤ b' ≤ b, the retraction of M_b restricted to M_{b'} is
homotopic relative to M_a to the composite of the retractions of M_{b'} onto M_{a'} and of
M_{a'} onto M_a (TauCeti.homotopicRel_sublevel_retraction_comp_sublevelInclusion).
Following Milnor, the deformation is the flow of a vector field X with df(X) = -1 on the band,
along which f decreases at unit speed. Since df(X) ≤ 0 everywhere, the time-(b - a) map of
this flow is a diffeomorphism of M carrying M_b onto M_a and the level f⁻¹(b) onto
f⁻¹(a) (TauCeti.exists_diffeomorph_sublevel); in particular M_b is homeomorphic to M_a
(TauCeti.exists_sublevelHomeomorph).
Following each point of the band down its flow line to the level a shows that the band itself is
a product: f⁻¹[a, b] is homeomorphic to f⁻¹(a) × [a, b] by a homeomorphism preserving the value
of f and fixing the level f⁻¹(a) (TauCeti.exists_bandHomeomorph).
Running such a flow for a longer time deforms the whole manifold, through maps along which f
never increases, so that M_b ends in the open sublevel {f < a}
(TauCeti.exists_ambientSublevelDeformation). Across a critical level c the same flow, built
away from a compact neighbourhood of the critical points, deforms {f ≤ c + ε} into
{f < c - ε} ∪ U for any open U containing the critical points of a compact band around c
(TauCeti.exists_criticalLevelDeformation). These critical points may be degenerate, and need
not be finite in number. Such ambient deformations are the steps of the Lusternik–Schnirelmann
bound on the number of critical points of a smooth function.
Main declarations #
TauCeti.exists_flow_apply_eq_sub_of_isCompact: a smooth flow along whichfnever increases, loweringfat unit speed near a compact set without critical points.TauCeti.exists_flow_apply_eq_sub: such a flow loweringfat unit speed through a compact band without critical points.TauCeti.exists_sublevelDeformationandTauCeti.exists_sublevelHomotopyEquiv: the deformation retraction ofM_bontoM_aand the resulting homotopy equivalence.TauCeti.homotopicRel_sublevel_retractionandTauCeti.homotopicRel_sublevel_retraction_comp_sublevelInclusion: retractions ofM_bontoM_aare unique up to homotopy relative toM_a, and so commute with restriction to smaller bands.TauCeti.exists_diffeomorph_sublevelandTauCeti.exists_sublevelHomeomorph: a diffeomorphism ofMcarryingM_bontoM_aandf⁻¹(b)ontof⁻¹(a), and the resulting homeomorphismM_b ≃ₜ M_a.TauCeti.exists_bandHomeomorph: the product structuref⁻¹[a, b] ≃ₜ f⁻¹(a) × [a, b]of the band.TauCeti.exists_ambientSublevelDeformation: a deformation ofMcarryingM_binto{f < a}.TauCeti.exists_criticalLevelDeformation: a deformation ofMcarrying{f ≤ c + ε}into{f < c - ε} ∪ U, for arbitrary, possibly degenerate, critical points inU.
References #
- J. Milnor, Morse Theory, Annals of Mathematics Studies 51, Princeton University Press, 1963, Theorem 3.1.
- J. Milnor, Lectures on the h-cobordism theorem, Princeton University Press, 1965, Theorem 3.4.
- O. Cornea, G. Lupton, J. Oprea and D. Tanré, Lusternik–Schnirelmann Category, Mathematical Surveys and Monographs 103, American Mathematical Society, 2003, Chapter 1.
A descending flow lowering f at unit speed near a compact set of regular points. If f
is smooth and the compact set K contains no critical point of f, there is a flow φ on M,
jointly smooth in time and space, along which f never increases, and an open neighbourhood U
of K such that f decreases at unit speed along every orbit segment lying in U: if
φ s x ∈ U for all s ∈ [r, t], then f (φ t x) = f (φ r x) - (t - r).
A unit-speed descending flow 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, there is a flow φ on M,
jointly smooth in time and space, along which f never increases, strictly decreases from every
point of the band, and decreases at unit speed through the band: if f x ≤ b, t ≥ 0 and
f x - t ≥ a, then f (φ t x) = f x - t.
Deformation of sublevels across a band without critical points (Milnor, Theorem 3.1). If
f is smooth, a ≤ b, and the band f⁻¹[a, b] is compact and contains no critical point of f,
then the sublevel {f ≤ a} is a strong deformation retract of {f ≤ b}: there is a retraction
r of {f ≤ b} onto {f ≤ a} and a homotopy from the inclusion of r to the identity fixing
{f ≤ a} pointwise.
The sublevels on either side of a band without critical points are homotopy equivalent.
Under the hypotheses of TauCeti.exists_sublevelDeformation, the inclusion of {f ≤ a} into
{f ≤ b} is a homotopy equivalence.
Retractions across a band without critical points are unique up to homotopy. Under the
hypotheses of TauCeti.exists_sublevelDeformation, any two retractions of {f ≤ b} onto
{f ≤ a} are homotopic relative to {f ≤ a}. In particular, the deformation retraction does not
depend, up to such a homotopy, on the descending vector field or the metric used to build it.
Retractions across nested bands commute up to homotopy. Let a ≤ a' ≤ b' ≤ b, with the
band f⁻¹[a, b'] compact and free of critical points. Restricting a retraction of {f ≤ b} onto
{f ≤ a} to {f ≤ b'} gives, up to homotopy relative to {f ≤ a}, the composite of a retraction
of {f ≤ b'} onto {f ≤ a'} with a retraction of {f ≤ a'} onto {f ≤ a}. Nothing is assumed
about the band f⁻¹[b', b].
Sublevels on either side of a band without critical points are diffeomorphic
(Milnor, Theorem 3.1). If f is smooth, a ≤ b, and the band f⁻¹[a, b] is compact and contains
no critical point of f, then there is a diffeomorphism Φ of M carrying the sublevel
{f ≤ b} onto {f ≤ a} and the level f⁻¹(b) onto f⁻¹(a).
Sublevels on either side of a band without critical points are homeomorphic
(Milnor, Theorem 3.1). Under the hypotheses of TauCeti.exists_diffeomorph_sublevel, there is a
homeomorphism from {f ≤ b} to {f ≤ a} carrying the level f⁻¹(b) onto f⁻¹(a).
A band without critical points is a product (Milnor, Lectures on the h-cobordism
theorem, Theorem 3.4). If f is smooth and the band f⁻¹[a, b] is compact and contains no
critical point of f, then there is a homeomorphism e : f⁻¹[a, b] ≃ₜ f⁻¹(a) × [a, b] whose
second component is the value of f and which is the identity on the level f⁻¹(a). In particular
each level f⁻¹(s), a ≤ s ≤ b, is carried onto f⁻¹(a) × {s}.
Deformation into an open lower sublevel 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, there is a
deformation Φ of the whole manifold M, starting at the identity, along which f never
increases and whose endpoint carries the closed sublevel {f ≤ b} into the open sublevel
{f < a}.
Deformation near a critical level. Let f be smooth, let the band f⁻¹[c - δ, c + δ],
δ > 0, be compact, and let U be an open set containing every critical point of f in this
band. Then for some 0 < ε < δ there is a deformation Φ of the whole manifold M, starting at
the identity, along which f never increases and whose endpoint carries the sublevel
{f ≤ c + ε} into {f < c - ε} ∪ U.
No nondegeneracy, finiteness or isolation of the critical points is assumed. When the critical
points of the band all lie on the level c, this is the deformation across a critical level used
to bound the Lusternik–Schnirelmann category of M by the number of critical points of f.