Morse functions are dense

On a boundaryless finite-dimensional Hausdorff manifold M, every C^n function, 2 ≤ n ≤ ∞, can be approximated, in the weak Whitney topology on C^n⟮I, M; 𝓘(ℝ), ℝ⟯, by C^n functions all of whose critical points in a prescribed compact set K are nondegenerate. On a compact manifold these functions form an open dense set, by the openness theorem TauCeti.isOpen_setOf_forall_mem_criticalPoints_isManifoldNondegenerateCriticalPoint, and hence a residual one. For smooth functions this says that Morse functions are dense; together with TauCeti.isOpen_setOf_isMorse they form an open dense set, and in particular every compact manifold carries a Morse function. On a compact manifold the weak Whitney topology is also the strong one, so this is the compact case of the Morse approximation theorem. The finite-order statement is the one used for genericity in Banach spaces of C^k functions: Morse functions in the sense of TauCeti.IsMorse are smooth, so for finite n the property is stated directly with TauCeti.IsManifoldNondegenerateCriticalPoint. The bound n ≥ 2 is needed for the openness, and the manifold is smooth, so that bump functions and charts are available at every order.

The proof is the classical one. Near a point c, choose a smooth bump function β at c and perturb a C^n function g to y ↦ g y - a (β y • φ y), where φ is the extended chart at c and a is a continuous linear functional. On the open set W where β = 1 inside the chart source, this is g - a ∘ φ, so in coordinates it is a linear perturbation of g ∘ φ.symm, and for a dense set of a it has only nondegenerate critical points there (TauCeti.dense_setOfPred_hasNondegenerateCriticalPointsOn_sub, which needs only C²). The perturbation is jointly C^n in a and y, hence continuous in a for the weak Whitney topology (ContMDiff.continuous_manifoldWeakWhitney), so small a keep it in any prescribed neighbourhood of g, and, by the openness of nondegeneracy on compact sets (TauCeti.eventually_forall_isManifoldNondegenerateCriticalPoint), keep its critical points nondegenerate on the compact set where those of g already were. Covering K by finitely many such W and perturbing one chart at a time gives a function whose critical points in all of K are nondegenerate.

Main results

• ContMDiffMap.exists_forall_isManifoldNondegenerateCriticalPoint_hasCompactSupport_sub: make the critical points of a C^n function in a compact set nondegenerate by a compactly supported perturbation inside a prescribed open set, keeping this property on an already treated compact set; ContMDiffMap.exists_isMorseOn_union_hasCompactSupport_sub is its smooth form.
• TauCeti.dense_setOf_forall_isManifoldNondegenerateCriticalPoint: for a compact set K, the C^n functions with only nondegenerate critical points in K are dense in the weak Whitney topology, and TauCeti.dense_setOf_isMorseOn is its smooth form.
• TauCeti.dense_setOf_forall_mem_criticalPoints_isManifoldNondegenerateCriticalPoint: on a compact manifold, the C^n functions with only nondegenerate critical points are dense.
• TauCeti.dense_setOf_isMorse: on a compact manifold, Morse functions are dense.
• TauCeti.exists_isMorse: a compact manifold carries a Morse function.

References

• M. Hirsch, Differential Topology, GTM 33, Springer, 1976, Chapter 6, §1, where Morse functions are shown to form a dense open set by local linear perturbations.
• V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, 1974, Chapter 1, §7.
• J. Milnor, Morse Theory, Annals of Mathematics Studies 51, Princeton University Press, 1963, §6, for the existence of Morse functions by embedding the manifold in Euclidean space.

theorem ContMDiffMap.exists_forall_isManifoldNondegenerateCriticalPoint_hasCompactSupport_sub {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] [FiniteDimensional ℝ E] [IsManifold I (↑⊤) M] [T2Space M] {n : ℕ∞} [IsManifold I (↑n) M] (hn : 2 ≤ n) (f : ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ ↑n) {K U A : Set M} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) (hA : IsCompact A) (hfA : ∀ x ∈ A, x ∈ TauCeti.criticalPoints I ⇑f → TauCeti.IsManifoldNondegenerateCriticalPoint I (⇑f) x) {𝒰 : Set (ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ ↑n)} (h𝒰 : 𝒰 ∈ nhds f) : ∃ g ∈ 𝒰, (∀ x ∈ A ∪ K, x ∈ TauCeti.criticalPoints I ⇑g → TauCeti.IsManifoldNondegenerateCriticalPoint I (⇑g) x) ∧ (HasCompactSupport fun (y : M) => g y - f y) ∧ (tsupport fun (y : M) => g y - f y) ⊆ U

For n ≥ 2, a C^n function can be perturbed, by an arbitrarily small weak Whitney perturbation with compact support inside any open neighbourhood U of a compact set K, to have only nondegenerate critical points in K. The perturbed function keeps this property on a specified compact set A on which the original function already had it. In particular it is unchanged on a neighbourhood of every point outside U.

theorem ContMDiffMap.exists_isMorseOn_union_hasCompactSupport_sub {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] [FiniteDimensional ℝ E] [IsManifold I (↑⊤) M] [T2Space M] (f : ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ ↑⊤) {K U A : Set M} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) (hA : IsCompact A) (hfA : TauCeti.IsMorseOn I (⇑f) A) {𝒰 : Set (ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ ↑⊤)} (h𝒰 : 𝒰 ∈ nhds f) : ∃ g ∈ 𝒰, TauCeti.IsMorseOn I (⇑g) (A ∪ K) ∧ (HasCompactSupport fun (y : M) => g y - f y) ∧ (tsupport fun (y : M) => g y - f y) ⊆ U

A smooth function can be made Morse on a compact set K by an arbitrarily small weak Whitney perturbation with compact support inside any open neighbourhood U of K. The perturbed function remains Morse on a specified compact set A on which it already was Morse. In particular it is unchanged on a neighbourhood of every point outside U.

theorem TauCeti.dense_setOf_forall_isManifoldNondegenerateCriticalPoint {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] [FiniteDimensional ℝ E] [IsManifold I (↑⊤) M] [T2Space M] {n : ℕ∞} [IsManifold I (↑n) M] (hn : 2 ≤ n) {K : Set M} (hK : IsCompact K) : Dense {f : ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ ↑n | ∀ x ∈ K, x ∈ criticalPoints I ⇑f → IsManifoldNondegenerateCriticalPoint I (⇑f) x}

Nondegenerate critical points on a compact set are dense. For a compact set K of a boundaryless finite-dimensional Hausdorff manifold and n ≥ 2, every C^n function is a limit, in the weak Whitney topology, of C^n functions all of whose critical points in K are nondegenerate.

theorem TauCeti.dense_setOf_forall_mem_criticalPoints_isManifoldNondegenerateCriticalPoint {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] [FiniteDimensional ℝ E] [IsManifold I (↑⊤) M] [T2Space M] {n : ℕ∞} [IsManifold I (↑n) M] [CompactSpace M] (hn : 2 ≤ n) : Dense {f : ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ ↑n | ∀ x ∈ criticalPoints I ⇑f, IsManifoldNondegenerateCriticalPoint I (⇑f) x}

Functions with only nondegenerate critical points are dense in the C^n functions, n ≥ 2, on a compact boundaryless finite-dimensional Hausdorff manifold, for the weak Whitney topology. With TauCeti.isOpen_setOf_forall_mem_criticalPoints_isManifoldNondegenerateCriticalPoint they form an open dense set, hence a residual one (residual_of_dense_open).

theorem TauCeti.dense_setOf_isMorseOn {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] [FiniteDimensional ℝ E] [IsManifold I (↑⊤) M] [T2Space M] {K : Set M} (hK : IsCompact K) : Dense {f : ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ ↑⊤ | IsMorseOn I (⇑f) K}

Morse functions on a compact set are dense. For a compact set K of a boundaryless finite-dimensional Hausdorff manifold, every smooth function is a limit, in the weak Whitney topology, of smooth functions which are Morse on K.

theorem TauCeti.dense_setOf_isMorse {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] [T2Space M] [CompactSpace M] : Dense {f : ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ ↑⊤ | IsMorse I ⇑f}

Morse functions are dense in the smooth functions on a compact boundaryless finite-dimensional Hausdorff manifold, for the weak Whitney topology.

theorem TauCeti.exists_isMorse {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] [T2Space M] [CompactSpace M] : ∃ (f : M → ℝ), IsMorse I f

Every compact manifold carries a Morse function: a compact boundaryless finite-dimensional Hausdorff manifold has a smooth function all of whose critical points are nondegenerate.