SUMS OF ISOLATED CRITICAL POINTS

Henry C. King

University of Maryland

Abstract. Suppose f_{i}: (X_{i}, x_{i}) → (R, 0) for i = 0, 1 are two functions with isolated critical points at x_{i}. Then the sum f_{0} +f_{1}: X_{0} ×X_{1} → R, namely (f_{0} +f_{1})(x_{0}, x_{1}) = f_{0}(x_{0}) + f_{1}(x_{1}), has an isolated critical point at (x_{0}, x_{1}). Assuming that the spaces X_{i} are well behaved, we give a local description of the behavior of the function f_{0} + f_{1}. As an application we give a short proof of a result in Morse theory on Whitney stratified sets.

If f: X → R is a continuous function on a topological space X, a point p ∈ X is called a regular point of f if there is an ϵ > 0, a neighborhood V of p in f^{−1}(f(p)) and an imbedding h: (−ϵ, ϵ) × V → X onto a neighborhood of p so that fh(u, v) = f(p) + u. Of course we want the imbedding to respect any structure on X in which we are interested. For example if X is a CS set (a natural collection of topological stratified sets, see [S]), we want V to be a CS set also and we want h to preserve strata. Notice that in the smooth manifold case the implicit function theorem implies that this notion of regular point is equivalent to the usual notion, namely that df_{p} be nonsingular. A point which is not regular is called a critical point of f. A point x ∈ X is an isolated critical point of f if there is a neighborhood U of x in X so that all points of U − x are regular points of f.

In this paper we will not dwell on CS sets too much, instead we will indicate proofs in the more familiar contexts of smooth manifolds or Whitney stratified sets where vector field techniques may be used. For those readers familiar with [S], the modifications in the proofs for CS sets will be clear, once the original proofs in [K2] and [K3] are referred to. Those not familiar with [S] may be happier ignoring the CS set case.

In [K1], [K2] and [K3] isolated critical points of functions were studied and characterized topologically. The case of maps between manifolds was done explicitly, although it was pointed out that the proofs applied to a much more general class of spaces, in particular Siebenmann’s CS sets. The main tool in these papers was a parameterized family of topological flows on the domain which was compatible with the function.

In particular, if X is a manifold or more generally a CS set and f: (X, x_{0}) → (R, 0) is a continuous function with an isolated critical at x_{0}, then in [K2] it was

Typeset by AMS-TEX

Supported in part by the Graduate Research Board, University of Maryland

shown there is a neighborhood C of x_{0} in X, a δ > 0 and a continuous flow Φ: U → C where U is a subset of C × R so that

(1) fΦ(x, t) = f(x) + t for all (x, t) ∈ U.

(2) Φ(x, 0) = x and Φ(Φ(x, t), s) = Φ(x, s + t) whenever these expressions are defined. (In other words, Φ is a flow.)

(3) C ⊂ f^{−1}([−δ, δ]).

(4) For each x ∈ C,

U ∩ x × R = \{ \begin{matrix}x × [−f(x), δ − f(x)] & if f(x) > 0 \\ x × [−δ − f(x), −f(x)] & if f(x) < 0 \\ x × [−δ, δ] & if f(x) = 0 and x ≠ x_{0} \\ x_{0} × 0 & if x = x_{0} \end{matrix}

The above flow Φ was called p in [K2]. We call the triple (C, δ, Φ) critical point data for f at x_{0}.

Let N = C ∩ f^{−1}(0). Note we have a deformation retraction φ: C → N given by φ(x) = Φ(x, −f(x)). We can actually enlarge the domain U of Φ while still satisfying 1), 2) and 3) by setting Φ(x, t) = Φ(φ(x), f(x) + t) if φ(x) ≠ x_{0}. Then Φ(x, t) is defined for all x ∈ [−δ − f(x), δ − f(x)] if φ(x) ≠ x_{0}. From now on we will use this enlarged Φ.

Figure 1: The flow Φ after enlarging U

It will be useful to say a few words about the construction of the neighborhood C. We start out with any compact neighborhood V of x_{0} in X. Let ∂V denote the frontier of V. We then find a neighborhood N′ of ∂V ∩ f^{−1}(0) in f^{−1}(0) − x_{0} and an imbedding τ: N′ × (−δ′, δ′) → X so that τ(x, 0) = x and fτ(x, t) = t. Pick δ ∈ (0, δ′) so that δ is less than the minimum of |f(x)| for x ∈ ∂V −τ(N′×(−δ′, δ′)). Then we can take C = τ(N′ × [−δ, δ]) ∪ (V ∩ f^{−1}([−δ, δ])).

If X and f are somehow smooth then the existence of Φ is easy to show, it is a flow obtained from integrating a vector field. For example, if X = R^{n} and f is smooth, we can integrate the vector field ∇f/|∇f|^{2} on X − x_{0}, except that we modify the vector field a little to be tangent to the curves τ(x × [−δ, δ]) for x ∈ N′ − C′. More generally, if X is a Whitney stratified set and f is smooth, we may integrate a controlled vector field 𝔳 so that df(𝔳) = d/dt, the standard unit vector field on R. Without smoothness one must give other arguments. For example if X is a CS set one can use the tools in [S] as was done in [K2].

Figure 2: The construction of C

We define Morse data for f as follows. Pick critical point data (C, δ, Φ) for f at x_{0}. Let Morse data be the pair (C, C ∩ f^{−1}(−δ)). Of course Morse data is not unique. The idea of Morse data is that if X is compact (or f is proper) and x_{0} is the only critical point of f in f^{−1}([−δ, δ]) then the pair (f^{−1}((−∞, δ]), f^{−1}((−∞, −δ])) is, up to excision and homotopy equivalence, the same as the Morse data. Moreover, f^{−1}((−∞, δ]) is homeomorphic to f^{−1}((−∞, −δ]) ∪ C. (We are assuming a few things above, in particular that X is a CS set or other well behaved space, C is compact and C ∩ f^{−1}(0) is collared along ∂C ∩ f^{−1}(0).) As shown in [K2], Morse data determines the function locally up to continuous change of coordinates.

Although Morse data is not unique, its algebraic topology is unique. Also it was shown in [K2] that the Morse data is unique if you fix C ∩ f^{−1}(0), even if you just fix it up to isotopy. The only uniqueness result we need here is the folowing:

Proposition 1. If f: (X, x_{0}) → (R, 0) has an isolated critical point at x_{0} and X is a CS set then for any perversity p̄ and any two Morse data (C, A) and (C′, A′) for f at x_{0} there is an isomorphism ι_{∗}: IHp̄ ; ∗(C′, A′) → IHp̄ ; ∗(C, A). Also, their ordinary homologies are isomorphic.

Proof. We must have two critical point data (C, δ, Φ) and (C′, δ′, Φ′) for f where A = C ∩f^{−1}(−δ) and A′ = C′∩f^{−1}(−δ′). Let φ: C → C ∩f^{−1}(0) and φ′: C′ → C′∩ f^{−1}(0) be the retractions φ(x) = Φ(x, −f(x)) and φ′(x) = Φ′(x, −f(x)). Note that if 0 < δ′′ ≤ δ then (C, A) is homeomorphic to (C ∩ f^{−1}([−δ′′, δ′′]), C ∩ f^{−1}(−δ′′)). For example, take the homeomorphism x ↦ Φ(x, (δ′′/δ − 1)f(x)). Thus the Morse data is independent of δ so we may as well assume that δ = δ′ and δ is small.

Pick any compact neighborhood N′′ of x_{0} in f^{−1}(0) ∩ C ∩ C′. If C′′ = φ′^{−1}(N′′) then (C′′, δ, Φ′′) is critical point data for f where Φ′′ is the restriction of Φ′. So we get Morse data (C′′, C′′ ∩ f^{−1}(−δ)). Consequently we can get arbitrarily small Morse data. Thus we may as well assume that C′ is compact and C′ ⊂ C.

At this point we can indicate why Morse data is unique if C ∩ f^{−1}(0) is isotopic to C′ ∩ f^{−1}(0). By the isotopy extension theorem, we know that (C − IntC′′) ∩ f^{−1}(0) is isomorphic to (C′ − IntC′′) ∩ f^{−1}(0). For δ small enough, (C − IntC′′) ∩ f^{−1}([−δ, δ]) and (C′ − IntC′′) ∩ f^{−1}([−δ, δ]) fiber over [−δ, δ]. But all bundles over [−δ, δ] are trivial so (C − IntC′′) ∩ f^{−1}([−δ, δ]) and (C′ − IntC′′) ∩ f^{−1}([−δ, δ]) are isomorphic. Thus C and C′ are obtained from C′′ in an isomorphic way, and hence are isomorphic. A more complete argument appears in [K2].

We will now show that the inclusion ι: (C′, A′) → (C, A) induces isomorphisms on homology and intersection homology with any perversity. Pick a compact neighborhood K of x_{0} in f^{−1}(0) ∩ Int C′. Pick a continuous function g: f^{−1}(0) → [0, 1] so that C − K ⊂ g^{−1}(1) and g^{−1}(0) is a neighborhood K′ of x_{0}. By choosing δ small enough, we may assume that φ^{−1}(K) ⊂ C′. Define h: C → C by h(x) = Φ(x, −gφ(x)(δ + f(x))). Note that the image of h is contained in C′ ∪ A.

Figure 3: The image of h

Pick any σ ∈ IHp̄ ; i(C, A). Let ι_{∗} : IHp̄ ; ∗(C′, A′) → IHp̄ ; ∗(C, A) be the map induced by inclusion. Pick a representative for σ, say a singular i-chain s (see [K4] for a definition of intersection homology in terms of singular chains). Let s′ be the singular i-chain obtained by composing all maps with h. Then s and s′ are homologous since there is a stratum preserving homotopy of h to the identity. Let s′′ be s′ except we eliminate all simplices which hit A − C′. Then if we chose a fine enough representative s for σ, we know s′ and s′′ will be homologous since all eliminated simplices have image in A. But s′′ represents a class in IHp̄ ; i(C′, A′) so we have shown ι_{∗} is onto.

Now let us show injectivity of ι_{∗}. If ι_{∗}(σ) = 0 then by the surjectivity argument above applied to (φ^{−1}(K′), φ^{−1}(K′) ∩ A) ⊂ (C′, A′) we may assume that σ has a representative with support in φ^{−1}(K′). Now use h to flatten any null homology of σ to one in (C′, A′). □

The main result of this paper is the following. A homology version in the manifold case appeared in [K1].

Theorem 2. Let X_{i} be CS sets and let f_{i}: (X_{i}, x_{i}) → (R, 0) have isolated critical points at x_{i} for i = 0, 1. Then f_{0} + f_{1} has an isolated critical point at (x_{0}, x_{1}). Furthermore, if (C_{i}, A_{i}) is Morse data for f_{i}, then some Morse data for f_{0} + f_{1} is isomorphic to (C_{0}, A_{0}) × (C_{1}, A_{1}) = (C_{0} × C_{1}, C_{0} × A_{1} ∪ A_{0} × C_{1}).

Proof. We have critical point data (C_{i}, δ_{i}, Φ_{i}) for the functions f_{i} at x_{i}. Pick δ > 0 so that 4δ ≤ δ_{i} for i = 0, 1. Let f denote f_{0} + f_{1}.

We first show that f has an isolated critical point at (x_{0}, x_{1}). Pick any (y_{0}, y_{1}) ∈ C_{0} × C_{1} − (x_{0}, x_{1}). Suppose y_{0} ≠ x_{0}. Then for a small neighborhood U of (y_{0}, y_{1}) in f^{−1}(f(y_{0}, y_{1})) and a small ϵ > 0 we have an imbedding h: U × [−ϵ, ϵ] → C_{0} × C_{1} so that fh(z, t) = f(y_{0}, y_{1}) + t for all z ∈ U and t ∈ [−ϵ, ϵ]. For example, take h((z_{0}, z_{1}), t) = (Φ_{0}(z_{0}, t), z_{1}). Thus (y_{0}, y_{1}) is not a critical point of f. If y_{0} = x_{0} then y_{1} ≠ x_{1} so a symmetric argument shows that (y_{0}, y_{1}) is not a critical point. So f has an isolated critical point at (x_{0}, x_{1}).

We now want to produce Morse data for f and show it is the product of the Morse data (C_{i}, A_{i}) for f_{i}.

Pick compact neighborhoods N′_{i} and N′′_{i} of x_{i} in N_{i} = C_{i} ∩ f_{i}^{−1}(0) so that N′_{i} ⊂ Int N′′_{i} and N′′_{i} ⊂ Int N_{i}. Let

C = { (z_{0}, z_{1}) ∈ C_{0} × C_{1} : |f_{0}(z_{0}) + f_{1}(z_{1})| ≤ δ and |f_{0}(z_{0}) − f_{1}(z_{1})| ≤ 4δ }
A = C ∩ f^{−1}(−δ)
N = C ∩ f^{−1}(0) = { (z_{0}, z_{1}) ∈ C_{0} × C_{1} : f_{0}(z_{0}) = −f_{1}(z_{1}) and |f_{0}(z_{0})| ≤ 2δ }
N′ = N − { (z_{0}, z_{1}) : φ_{i}(z_{i}) ∈ N′_{i} and |f_{0}(z_{0})| ≤ δ }

We will find a collaring of N′ which will show that (C, A) is in fact Morse data for f.

Consider the rectangle Q = { (s, t) ∈ R^{2} : |s + t| ≤ δ and |s − t| ≤ 4δ }. Note that C = C_{0} × C_{1} ∩ f^{−1}(Q). Let λ: [−2δ, 2δ] × [−δ, δ] × [0, 1] → Q × [0, 1] be a homeomorphism so that:

(1) λ(s, t, u) = ((λ_{0}(s, t, u), λ_{1}(s, t, u), u) for some functions λ_{i},

(2) λ_{0}(s, t, u) + λ_{1}(s, t, u) = t,

(3) λ(s, 0, u) = (s, −s, u),

(4) λ(s, t, u) = (s + t/2, −s + t/2, u) if |s| ≥ δ,

(5) λ(0, t, 0) = (0, t, 0) and λ(0, t, 1) = (t, 0, 1). Think of λ as a parameterized family of homeomorphisms of [−2δ, 2δ]×[−δ, δ] with Q. The images of lines s = constant for u = 0, 1 are indicated in figure 4.

Figure 4: The map λ

Pick continuous maps γ_{i}: N_{i} → [0, 1] so that x_{i} = γ_{i}^{−1}(0). Define β_{i}: N′ × [−δ, δ] → R and γ: C_{0} × C_{1} − φ_{0}^{−1}(x_{0}) × φ_{1}^{−1}(x_{1}) → [0, 1] by

γ(z_{0}, z_{1}) = γ_{0}φ_{0}(z_{0})/(γ_{0}φ_{0}(z_{0}) + γ_{1}φ_{1}(z_{1}))
β_{i}(z_{0}, z_{1}, t) = \{ \begin{matrix}t/2 & if φ_{0}(z_{0}) ∈ N′_{0} and φ_{1}(z_{1}) ∈ N′_{1} \\ λ_{i}(f_{0}(z_{0}), t, γ(z_{0}, z_{1})) − f_{i}(z_{i}) & otherwise \end{matrix}

The maps β_{i} are continuous because if φ_{j}(z_{j}) ∈ N′_{j} for j = 0, 1 then |f_{0}(z_{0})| > δ so λ_{i}(f_{0}(z_{0}), t, γ(z_{0}, z_{1})) = (−1)^{i}f_{0}(z_{0}) + t/2 = f_{i}(z_{i}) + t/2.

We have a collaring map τ: N′ × [−δ, δ] → C given by

τ(z_{0}, z_{1}, t) = (Φ_{0}(z_{0}, β_{0}(z_{0}, z_{1}, t)), Φ_{1}(z_{1}, β_{1}(z_{0}, z_{1}, t))) .

This collaring gives us (C, A) as Morse data. For example, if V ⊂ C_{0} × C_{1} is the set of points where |f_{0}(z_{0})| + |f_{1}(z_{1})| ≤ 3δ and φ_{i}(z_{i}) ∈ N′′_{i} then V is a neighborhood of (x_{0}, x_{1}), N′ is a neighborhood of ∂V ∩ f^{−1}(0) and C = τ(N′ × [−δ, δ])∪(V ∩f^{−1}([−δ, δ])). To assist the reader who wishes to verify this, note that if (z_{0}, z_{1}) = τ(w_{0}, w_{1}, t) then φ_{i}(z_{i}) = φ_{i}(w_{i}) so γ(z_{0}, z_{1}) = γ(w_{0}, w_{1}) and w_{i} = Φ_{i}(z_{i}, (−1)^{i}s − f_{i}(z_{i})) where (s, f(z_{0}, z_{1}), γ(z_{0}, z_{1})) = λ^{−1}(f_{0}(z_{0}), f_{1}(z_{1}), γ(z_{0}, z_{1})) (assuming γ is defined).

So it remains to show that (C, A) = (C_{0}, A_{0}) × (C_{1}, A_{1}). Let Q′ be the lower side of Q, namely Q′ = { (s, t) ∈ Q : s + t = −δ }. Let κ: [−δ_{0}, δ_{0}] × [−δ_{1}, δ_{1}] → Q be a homeomorphism so that

κ^{−1}(Q′) = −δ_{0} × [−δ_{1}, δ_{1}] ∪ [−δ_{0}, δ_{0}] × −δ_{1}

and so that κ_{0}(0, t) = 0 and κ_{1}(t, 0) = 0 for all t where κ_{0} and κ_{1} are the coordinates of κ. We have a homeomorphism h: (C_{0}, A_{0}) × (C_{1}, A_{1}) → (C, A) given by

h(z_{0}, z_{1}) = (Φ_{0}(z_{0}, κ_{0}(f_{0}(z_{0}), f_{1}(z_{1})) − f_{0}(z_{0})), Φ_{1}(z_{1}, κ_{1}(f_{0}(z_{0}), f_{1}(z_{1})) − f_{1}(z_{1}))) .

The inverse of h is given by a similar formula where we replace κ_{i} by the i-th coordinate of κ^{−1}. □

Before giving an application of Theorem 2 to Morse theory on Whitney stratified sets, we recall a result from [K3]. Suppose f_{z}: (X, x_{0}) → (R, 0), z ∈ Z is a family of functions. Suppose that x_{0} is an isolated critical point of each f_{z} and furthermore is ‘uniformly isolated’, i.e., for some neighborhood B of x_{0}, the map F: (B−x_{0})×Z → R × Z given by F (x, z) = (f_{z}(x), z) has no critical points. In the terminology of [K3], there is no coalescing. Note that the definition of no coalescing given in [K3] is incorrect and the above definition should be used. Of course the two definitions are the same in the smooth category, but are probably not the same in topological categories.

Theorem 3. Let f_{z}: (X, x_{0}) → (R, 0) for z ∈ Z be a continuous family of functions with uniformly isolated critical points, where X is a CS set. Then f_{z} is topologically trivial if and only if f_{z}^{−1}(0) is topologically trivial. In other words, for each z_{0} ∈ Z the following are equivalent:

(1) There are neighborhoods U of x_{0} in X and W of z_{0} in Z and a family of imbeddings h_{z}: (U, x_{0}) → (X, x_{0}) onto neighborhoods of x_{0} so that f_{z}h_{z} = f_{z_{0}} for all z ∈ W.

(2) There are neighborhoods V of x_{0} in f_{z_{0}}^{−1}(0) and W of z_{0} in Z and a family of imbeddings g_{z}: (V, x_{0}) → (f_{z}^{−1}(0), x_{0}) onto neighborhoods of x_{0} for z ∈ W.

Proof. This was essentially proven in Theorem 1 of [K3] in the case where X = R^{n} and the proof for X a CS set is exactly the same. In [K3], (2) was a little stronger in that it was assummed that g_{z} extended to a family of imbeddings of a neighborhood in X, but this was not really neccessary, it only simplified the notation in the proof a little.

Note (1) easily implies (2). We will indicate a proof that (2) implies (1) in the special case where (Z, z_{0}) = (R^{k}, 0), where X × Z is given a Whitney stratification and where F: X ×Z → R×Z and G: V ×Z → F^{−1}(0×Z) and G^{−1}: F^{−1}(0×Z) → V ×Z are smooth functions on each stratum off of x_{0}×Z where F (x, z) = (f_{z}(x), z), G(x, z) = (g_{z}(x), z) and G^{−1}(x, z) = (g_{z}^{−1}(x), z). Furthermore, we ask that G be controlled in the following sense: We may pick a controlled tubular neighborhood system (π_{U}, ρ_{U}) for (X − x_{0}) × Z so that if π: X × Z → Z is projection then π ∘ π_{U} = π and also F π_{U}(x) = (0, π(x)) if x ∈ F^{−1}(0 × Z). Then we ask that π_{U}G(x, z) = G(π_{U}(x), z) and ρ_{U}G(x, z) = ρ_{U}(x) if (x, z) ∈ V ×Z and (x, z) is near the stratum U. This is the only case we need for the rest of this paper.

For i = 1, 2, …, k, let 𝔴_{i} be the vector field 0 × ∂/∂z_{i} on R × Z. Pick controlled vector fields 𝔳_{i} on (U′ − x_{0}) × Z for a small neighborhood U′ of x_{0} in X so that dF (𝔳_{i}) = 𝔴_{i} and 𝔳_{i}(x, z) = dG(0 × ∂/∂z_{i}) if f_{z}(x) = 0. For each w ∈ R^{k}, let Ψ_{w} denote the flow associated to the vector field ∑ w_{i}𝔳_{i}. So

F Ψ_{w}((x, z), t) = (f_{z}(x), z + tw)
Ψ_{w}((g_{z}(x), z), t) = (g_{z+tw}(x), z + tw) if x ∈ V − x_{0}

Note that Ψ_{w} cannot flow into or out of x_{0}×Z. To see this, suppose that we have (x_{0}, z′) ∈ lim_{t→s} Ψ_{w}((x, z), t) where lim_{t→s} stands for the set of limiting points as t → s. Then

(0, z′) = F (x_{0}, z′) ∈ lim_{t→s} F Ψ_{w}((x, z), t) = lim_{t→s}(f_{z}(x), z + tw) = {(f_{z}(x), z + sw)}

so f_{z}(x) = 0, but then Ψ_{w}((x, z), t) = (g_{z+tw}g_{z}^{−1}(x), z + tw) so

(x_{0}, z′) ∈ lim_{t→s} Ψ_{w}((x, z), t) = {(g_{z+sw}g_{z}^{−1}(x), z + sw)}

a contradiction. Thus we may continuously extend Ψ_{w} by setting

Ψ_{w}((x_{0}, z), t) = (x_{0}, z + tw).

Pick a compact neighborhood U of x_{0} in Int U′. Let π: X × Z → X be projection. Define h_{z}(x) = πΨ_{z}((x, 0), 1) for x ∈ U, i.e., flow along to time one and then project. This is defined for all small enough z, since Ψ_{z}(w, t) = Ψ_{tz}(w, 1) by uniqueness of flows. It also has inverse h_{z}^{−1}(x) = πΨ_{z}((x, z), −1). Note that f_{z}h_{z}(x) = f_{0}(x) for all x ∈ U so we are done. □

We now give an application of Theorem 2 to Morse theory on Whitney stratified sets. For basic results on Whitney stratified sets we refer the reader to [GWPL].

Let Z ⊂ M be a Whitney stratified set ina smooth manifold M, let S be a stratum of Z, p a point in S and let f: M → R be a smooth function. We say f is normally nice at p if df_{p}(Q) ≠ 0 for all limits Q of tangent spaces to strata of Z − S. We say p is a critical point of f|_{Z} if df_{p}(T_{p}S) = 0. Otherwise we say p is a regular point of f|_{Z}. This is equivalent to our previous definition in this case by Thom’s isotopy Theorems.

Lemma 4. f is normally nice at any regular point.

Proof. If Q is the limit of tangent spaces of strata of Z − S then the Whitney conditions state that T_{p}S ⊂ Q. But df_{p}(T_{p}S) ≠ 0, hence df_{p}(Q) ≠ 0. □

Lemma 5. If f is normally nice at p then all nearby points of Z − S are regular points of f|_{Z}.

Proof. We know that df_{p}(Q) ≠ 0 for all limits Q of tangent spaces to any stratum S′ of Z − S. Hence for q ∈ S′ near p, df_{q}(T_{q}S′) ≠ 0 also. □

Lemma 6. If f is normally nice at p ∈ S, then f is normally nice at all q ∈ V where V is some neighborhood of p in Z.

Proof. Suppose not. Then there is a sequence p_{i} → p and tangent space limits Q_{i} so df_{p_{i}}(Q_{i}) = 0. By taking a subsequence, we may as well assume the Q_{i}’s all have the same dimension and the sequence Q_{i} approaches some limit Q. Since Q is a limit of limits of tangent spaces to strata of Z − S, it is itself a limit of tangent spaces of strata of Z − S. Hence df_{p}(Q) ≠ 0. Pick a v ∈ Q such that df_{p}(v) ≠ 0. Pick a sequence v_{i} ∈ Q_{i} such that v_{i} → v. Then 0 ≠ df_{p}(v) = lim(df_{p_{i}}(v_{i})) = lim(0) = 0, a contradiction. □

Lemma 7. Suppose f is normally nice at p and f^{−1}f(p) ∩ S is a smooth manifold near p. Then f^{−1}f(p) ∩ U is a Whitney stratified set for some neighborhood U of p in Z.

Proof. By Lemma 5, f|_{Z} has no critical points near p on any stratum of Z − S. Hence Y = f^{−1}f(p) ∩ U′ is a stratified set for some neighborhood U′ of p in Z. The strata of Y are just intersections of strata of Z with Y. By Lemma 6 we know f is normally nice at all points in some neighborhood of p, hence it suffices to show that Y satisfies the Whitney conditions at p, since all points in a neighborhood of p also satisfy the hypotheses of this lemma. We may as well assume M = R^{n}. So take any stratum S′ of Z which has S in its closure. Now take any sequences x_{i} ∈ S′ ∩ Y and y_{i} ∈ S ∩ Y so that x_{i} → p, y_{i} → p, T_{x_{i}}(S′ ∩ Y) → τ and the sequence of lines through x_{i} and y_{i} approaches some line l. By taking a subsequence, we may also assume that T_{x_{i}}S′ → τ′. Since f: M → R has a regular value at p, we know that f^{−1}f(p) is a manifold near p and hence l is tangent to this manifold and is thus in ker(df_{p}). Note that T_{x_{i}}(S′ ∩ Y) = T_{x_{i}}S′ ∩ ker(df_{x_{i}}) so τ ⊂ lim(T_{x_{i}}S′) = τ′ and τ ⊂ lim(ker(df_{x_{i}})) = ker(df_{p}). But df_{p}(τ′) ≠ 0 by normal nicety, so dim(τ′ ∩ ker(df_{p})) = dim(τ′) − 1. But dim(τ) = dim(τ′) − 1, so τ = τ′ ∩ ker(df_{p}). We know that T_{p}S ⊂ τ′ and l ⊂ τ′ by the Whitney conditions on Z. Hence

T_{p}(S ∩ Y) = T_{p}(S ∩ f^{−1}f(p)) ⊂ T_{p}S ∩ ker(df_{p}) ⊂ τ′ ∩ ker(df_{p}) = τ.

Also l ⊂ τ′ ∩ ker(df_{p}) = τ so the Whitney conditions are satisfied. □

Pick a submanifold N of M which is transverse to all strata of Z so that N ∩ S = {p}. Then we say N ∩ Z is a normal slice at p. One standard result on Whitney stratified sets is that the germ at p of a normal slice is unique up to homeomorphism. Figure 5 shows a normal slice for Z = { (x, y, z) : x^{2} = z^{3} }, S = { (x, y, z) : x = z = 0 } and N = { (x, y, z) : y = 0 and x^{2} + z^{2} < 1 }. The slice is the cusp N ∩ Z.

Note that any function of the form (x, y, z) ↦ z + g(y) is normally nice for this example. We can think of such a function as the sum of two functions, y ↦ g(y)

Figure 5: A normal slice

on S and (x, z) ↦ z on the normal slice. Theorem 8 shows that any normally nice function on a Whitney stratified set Z has a similar separation of variables.

Theorem 8. If f is normally nice at p then there is an imbedding μ: V × X → Z where V is a neighborhood of p in S and X is a normal slice at p so that fμ(v, x) = f(v) + f(x) − f(p) for all v ∈ V and x ∈ X.

Proof. Let π: T → S be a tubular neighborhood of S in M. By the theory of Whitney stratified spaces, after perhaps shrinking T there is an imbedding θ: U × X′ → Z onto a neighborhood of p where X′ = Z ∩ π^{−1}(p), U is a neighborhood of p in S, πθ(u, x) = u for all u ∈ U and x ∈ X′ (and thus θ(u, p) = u for all u ∈ U) and θ(p, x) = x for all x ∈ X′.

If p is a regular point of f|_{S} then we couild have chosen π so that fπ = f near p. Then we may let V = U and μ = θ since

fθ(y, x) = fπθ(y, x) = f(y) = f(y)+f(p)−f(p) = f(y)+fπ(x)−f(p) = f(y)+f(x)−f(p).

So we may suppose p is a critical point of f|_{S}. Let g: T → R be defined by g = f − fπ. Note that g is normally nice at p since if Q is a limit of tangent spaces of Z − S, then d(fπ)_{p}(Q) = df_{p}(dπ_{p}(Q)) ⊂ df_{p}(T_{p}S) = 0. Thus dg_{p}(Q) = d(f − fπ)_{p}(Q) = df_{p}(Q) ≠ 0.

Let g^{u}: X′ → R, u ∈ U be the family of functions defined by g^{u}(x) = gθ(u, x). We claim that this family has no coalescing of critical points. It suffices to show that there are no strata S′ of Z − S and sequences p_{i} → p so that p_{i} ∈ S′ and T_{p_{i}}(S′∩π^{−1}π(p_{i})) ⊂ ker(dg_{p_{i}}). If such a sequence exists, after taking a subsequence we can assume that T_{p_{i}}S′ → Q for some Q. By the Whitney conditions, we know that T_{p}S ⊂ Q. But T_{p}M = T_{p}S ⊕ T_{p}(π^{−1}(p)) so Q is transverse to T_{p}(π^{−1}(p)) and thus dim(Q ∩ T_{p}(π^{−1}(p))) = dim(Q) − dim(S). But lim(T_{p_{i}}(S′ ∩ π^{−1}π(p_{i}))) ⊂ Q ∩ T_{p}(π^{−1}(p)) so for dimension reasons T_{p_{i}}(S′ ∩ π^{−1}π(p_{i})) → Q ∩ T_{p}(π^{−1}(p)). So dg_{p}(Q ∩ T_{p}(π^{−1}(p))) = 0. But g vanishes on S so dg_{p}(Q) = dg_{p}((Q ∩ T_{p}(π^{−1}(p))) + T_{p}S) = 0 which contradicts the normal nicety of g. So the family has no coalescing.

Now by Lemma 7 we know that g^{−1}(0) is Whitney stratified, so after perhaps shrinking U there is an imbedding φ: U × Y → g^{−1}(0) onto a neighborhood of p in g^{−1}(0) so that πφ(u, y) = u, where Y is a neighborhood of p in X′ ∩ g^{−1}(0). Thus by Theorem 3 we have a family of imbeddings h^{u}: X → X′, u ∈ V where V is a neighborhood of p in U, X is a neighborhood of p in X′, g^{p} = g^{u}h^{u} for all u ∈ V and the image of each h^{u} is a neighborhood of p in X′. Define μ: V × X → Z by μ(u, x) = θ(u, h^{u}(x)). Note that πμ(u, x) = πθ(u, h^{u}(x)) = u. Also

g^{p}(x) = gθ(p, x) = g(x) = f(x) − fπ(x) = f(x) − f(p)

for all x ∈ X. Thus

fμ(u, x) = gμ(u, x) + fπμ(u, x) = gθ(u, h^{u}(x)) + f(u)
= g^{u}h^{u}(x) + f(u) = g^{p}(x) + f(u) = f(x) − f(p) + f(u). □

Theorem 9. If f is normally nice at p then Morse data for f|_{Z} is the product of Morse data for f|_{S} and f|_{X} where X is a normal slice at p.

Proof. For convenience, assume f(p) = 0. By Theorem 8, f|_{Z} can be written locally as the sum of f|_{S} and f|_{X} for a normal slice X. The result now follows from Theorem 2. □

In [GM] a similar result was proven, which they call the Main Theorem. Extra assumptions were made about the map f which among other things implied that there is a canonical choice of Morse data, namely (B ∩ f^{−1}([f(p) − δ, f(p) + δ]), B ∩ f^{−1}(f(p) − δ)) where B is a sufficiently small Reimannian ball about p. Then in [GM] it was actually shown that the canonical Morse data is a product of the canonical tangential and normal Morse data. It is not too hard for us to obtain this result also and we will give the proof below. However, since all one really needs in [GM] is the intersection homology of the Morse data, in view of Proposition 1 we have already essentially given a (much shorter) proof of the Main Theorem of [GM].

The reader who wishes to recover the full Main Theorem of [GM] may do so as follows. We may assume that Z ⊂ R^{k} × R^{n} so that the stratum S is R^{k} × 0 and f is defined on all of R^{k} × R^{n}. Let π: R^{k} × R^{n} → R^{k} be projection. Let f_{0} = f ∘ π and f_{1} = f − f_{0}. When convenient we will identify R^{k} and R^{n} with R^{k} × 0 and 0 × R^{n}. For example f_{0}(x) will mean f_{0}(x, 0).

We let (C_{0}, A_{0}) and (C_{1}, A_{1}) be canonical tangential and normal Morse data, say C_{0} = B^{k} × 0 ∩ f^{−1}([−δ_{0}, δ_{0}]) and C_{1} = Z ∩ 0 × B^{n} ∩ f^{−1}([−δ_{1}, δ_{1}]) where B^{k} and B^{n} are small balls. After going through all the argument in Theorem 2, while being careful to make the controlled vector fields in the proofs of Theorems 3 and 8 tangent to B^{k}×∂B^{n}, we obtain Morse data (C, A) isomorphic to (C_{0}, A_{0})×(C_{1}, A_{1}) where C = Z ∩B^{k} ×B^{n} ∩f^{−1}([−δ, δ])∩K and K = { (x, y) ∈ B^{k} ×B^{n} : |f_{0}(x, y)− f_{1}(x, y)| ≤ 4δ }. We must show that (C, A) is isomorphic to the canonical Morse data.

By [K2] it is only necessary to show that C ∩ f^{−1}(0) is isotopic in Z ∩ f^{−1}(0) to B ∩ Z ∩ f^{−1}(0) where B is a small ball, c.f., the proof of Proposition 1. We find this isotopy by exhibiting a controlled vector field 𝔳 on C ∩ f^{−1}(0) so that 𝔳 points inward on ∂C ∩ f^{−1}(0) and dr(𝔳) < 0 where r is the squared distance function. Note that C ∩ f^{−1}(0) = Z ∩ f^{−1}(0) ∩ L × B^{n} where L = { x ∈ B^{k} : |f(x, 0)| ≤ 2δ }. Of course L is homeomorphic to C_{0}.

Define r_{0}(x, y) = |x|^{2}, r_{1}(x, y) = |y|^{2} and r = r_{0} + r_{1}. Before constructing 𝔳 we will construct a useful vector field.

Lemma 10. If B^{n} and δ are chosen small enough, there is a controlled vector field 𝔴 defined on a neighborhood of C ∩ B^{k} × ∂B^{n} so that dπ(𝔴) = 0, df_{1}(𝔴) = 0 and dr_{1}(𝔴) < 0.

Proof. By Lemma 7 we know that Z′ = Z ∩ f_{1}^{−1}(0) is a Whitney stratified set. So we may pick an ϵ > 0 so that (π, r_{1}) is a submersion on each stratum of Z′ ∩ B^{k} × r_{1}^{−1}((0, 2ϵ)). Pick B^{n} with radius \sqrt{ϵ}. Then for any x ∈ B^{k}, we know that (x, 0, ϵ) is a regular value of (π, f_{1}, r_{1}) restricted to any stratum of Z. Then if δ is small enough, B^{k} × [−2δ, 2δ] × ϵ are all regular values of (π, f_{1}, r_{1}) restricted to any stratum of Z. So 𝔴 may be obtained by pulling back the vector field (0, 0, −1) on B^{k} × R × R. □

We now show that if f_{0} is a nice enough function, then we can find a vector field 𝔳 as above, and hence (C, A) is isomorphic to the canonical Morse data. For example if Z is an algebraic set or even a subanalytic set and f is analytic then the curve selection lemma easily implies hypotheses (1) and (2) below. By Lemma 12 below, the nondepravity condition in [GM] implies the hypotheses (1) and (2), so the whole of the Main Theorem of [GM] is proven.

Lemma 11. Suppose that:

(1) For all strata T of C ∩f^{−1}(0)−(0, 0), the restriction of r to T has no critical points. (Which is what you need to make sure f has canonical Morse data.)

(2) For all strata T of C ∩ f^{−1}(0) and (x, y) ∈ T with f_{0}(x) ≠ 0 we know that f_{0}(x)df_{0}|_{T} is not a negative multiple of dr|_{T} at (x, y).

Then if B^{n} and δ are small enough, there is a controlled vector field 𝔳 on C ∩ f^{−1}(0) − (0, 0) so that df(𝔳) = 0, dr(𝔳) < 0 and 𝔳 points inward on ∂C ∩ f^{−1}(0). Hence (C, A) is the canonical Morse data.

Proof. It suffices to construct 𝔳 pointwise. This is because by Lemma 4.11 of [Sh] we may then extend it locally to a continuous controlled vector field, which by continuity still satisfies dr(𝔳) < 0 and points inward on ∂C ∩ f^{−1}(0). We may then piece together these local vector fields with a partition of unity. Of course the partition of unity we use must be compatible with the tubular data for Z in order for the resulting vector field to be controlled. In particular, if φ is one of the partition functions and x is close to a stratum T then we want φ(x) = φπ_{T} (x) where π_{T} is the tubular projection to T. This is easy to arrange.

Note also that it is only really necessary on the boundary to find 𝔳 so that dr(𝔳) < 0 and 𝔳 either points inward or is tangent to ∂C ∩ f^{−1}(0). This is because if it is tangent, we need only add a small inward pointing vector to 𝔳 and it will point inward and still satisfy dr(𝔳) < 0.

Now pick any point (x, y) ∈ C ∩ f^{−1}(0). Let T be the tangent space to the stratum of C ∩ f^{−1}(0) containing (x, y) and let T′ be the tangent space to the stratum of C containing (x, y). So T = T′ ∩ ker(df). We are done if we can find a vector 𝔳 ∈ T so that dr(𝔳) < 0 and so that if (x, y) ∈ ∂C then 𝔳 points inward or is tangent to ∂C. By (1) we can certainly do this if (x, y) ∉ ∂C, so we may as well assume that (x, y) ∈ ∂C ∩ f^{−1}(0).

Suppose first that y ∈ ∂B^{n}. Then we may let 𝔳 be the 𝔴 of Lemma 10. Then dr(𝔴) = dr ∘ dπ(𝔴) + dr_{1}(𝔴) < 0. Also 𝔴 points inward or is tangent to ∂C since dr(𝔴) < 0. (It will be tangent if x ∈ ∂B^{k} or f_{0}(x) = ±2δ.) So we are done in this case.

So now suppose that x ∉ ∂B^{k} and y ∉ ∂B^{n}, so we have f_{0}(x) = ±2δ and f_{1}(x, y) = ∓2δ. If df_{0}|_{T} and dr|_{T} are linearly independent, there is a 𝔳 ∈ T so that df_{0}(𝔳) = −f(x, 0) and dr(𝔳) = −1. If they are not, we may take any vector 𝔳 ∈ T with dr(𝔳) < 0 and (2) implies that f_{0}(x)df_{0}(𝔳) ≤ 0 and hence that 𝔳 points inward or is tangent to ∂C.

So the remaining case is when y ∉ ∂B^{n} but x ∈ ∂B^{k}. We claim that if we had chosen B^{n} small enough then either d(r, r_{0}, f_{1})|_{T} has rank 3 or else d(r_{0}, f_{1}) has rank 2 and dr|_{T} = αdr_{0}|_{T} +βdf_{1}|_{T} with α > 0. Given this claim, in the rank 3 case we may then choose a 𝔳 ∈ T so that d(r, r_{0}, f_{1})(𝔳) = (−1, −1, 0) and in the other case we may choose a 𝔳 ∈ T so that d(r_{0}, f_{1}) = (−1, 0) and then dr(𝔳) = −α < 0.

So it only remains to prove the claim. If the claim is not true then there is a sequence of points (x_{i}, y_{i}) ∈ f^{−1}(0) so that x_{i} ∈ ∂B_{k}, (x_{i}, y_{i}) → (x, 0) for some x ∈ ∂B^{k} ∩f^{−1}(0) and if T_{i} is the tangent space to the stratum of C ∩f^{−1}(0) containing (x_{i}, y_{i}) then either d(r_{0}, f_{1})|_{T_{i}} has rank < 2 or dr|_{T_{i}} = α_{i}dr_{0}|_{T_{i}} +β_{i}df_{1}|_{T_{i}} with α_{i} ≤ 0. Let T′_{i} be the tangent space to the stratum of C containing (x_{i}, y_{i}). Then after taking a subsequence, we may assume that T_{i} → Q and T′_{i} → Q′ for some subspaces Q and Q′ of R^{k} × R^{n}. Let τ_{i}, τ and τ′ be the matrices of orthogonal projection to T_{i}, Q and Q′ respectively. Since f_{1}|_{Z} is normally nice we know that df_{1}(Q′) ≠ 0, which means that 𝔴_{1} = τ′(∇f_{1}(x)) ≠ 0. Note that 𝔴_{0} = τ′(∇f_{0}(x)) = ∇f_{0}(x) ≠ 0 since ∇f_{0} ∈ R^{k} × 0 ⊂ Q′ by the Whitney conditions. Since Q = Q′ ∩ ker df we know that if df_{1}(Q) = 0 then 𝔴_{1} = κτ′(∇f(x)) = κ(𝔴_{1} + 𝔴_{0}) for some κ. But ∇f_{1}(z) ∈ 0 × ℝ^{n} for any z ∈ R^{k} × 0 since f_{1}|_{ℝ^{k}×0} = 0. So for any z ∈ R^{k} × 0,

z · 𝔴_{1} = z · τ′(∇f_{1}) = τ′(z) · ∇f_{1} = z · ∇f_{1} = 0.

So 𝔴_{0} · 𝔴_{1} = 0 and hence 0 = κ|𝔴_{0}|^{2} ≠ 0, a contradiction. So df_{1}(Q) ≠ 0 and thus τ(∇f_{1}(x)) ≠ 0.

If τ(∇r_{0}(x, 0)) = λτ(∇f_{1}(x)), then ∇r_{0}(x) = λ𝔴_{1} + μ(𝔴_{0} + 𝔴_{1}) for some μ, but then 0 = ∇r_{0} · 𝔴_{1} = (λ + μ)|𝔴_{1}|^{2} so λ + μ = 0 and ∇r_{0} = μ𝔴_{0}. But (1) implies that ∇r_{0} and ∇f_{0} are linearly independent at x, a contradiction. So τ(∇r_{0}(x)) and τ(∇f_{1}(x)) are linearly independent. Since τ_{i} → τ we know that for large enough i, τ_{i}(∇r_{0}(x_{i}, y_{i})) and τ_{i}(∇f_{1}(x_{i}, y_{i})) are linearly independent so d(r_{0}, f_{1})|_{T_{i}} has rank 2. We also have τ(∇r(x)) = 1·τ(∇r_{0}(x))+0·τ(∇f_{1}(x)). Hence α_{i} → 1 and β_{i} → 0 which contradicts α_{i} ≤ 0. □

Lemma 12. If f has a nondepraved critical point at (0, 0) then it satisfies the hypotheses of Lemma 11, hence (C, A) is its canonical Morse data.

Proof. It was proven in [GM2] that f satisfies hypothesis (1).

For convenience, let h: R^{k} → R be the function h(x) = f(x, 0) = f_{0}(x, 0). We assume that ∇h(0) = 0. Let ρ: R^{n} → R be the last coordinate function. Since df_{1} ≠ 0 at (0, 0) and f_{1}(R^{k} × 0) = 0, we may assume after a smooth coordinate change that f_{1}(x, y) = ρ(y).

Suppose f does not satisfy hypothesis (2). Then there is a stratum T′ of Z and a sequence of points (x_{i}, y_{i}) ∈ T′∩f^{−1}(0) so that h(x_{i}) ≠ 0 and if T = T′∩f^{−1}(0) then h(x_{i})df_{0}|_{T} is a negative multiple of dr|_{T} at (x_{i}, y_{i}). We can not have y_{i} = 0 since then 0 = f(x_{i}, y_{i}) = h(x_{i}) ≠ 0. We can not have x_{i} = 0 since then 0 = h(x_{i})df_{0}|_{T} is not a negative multiple of dr|_{T}. So by taking a subsequence, we may assume that y_{i}/|y_{i}| → y and x_{i}/|x_{i}| → x for some unit vectors y ∈ R^{n} and x ∈ R^{k}.

We may assume the tangent spaces to T′ and T converge to Q′ and Q. As in the proof of Lemma 7, we know that Q = Q′ ∩ ker(df) = Q′ ∩ R^{k} × ker ρ. By the Whitney conditions we know that R^{k} × 0 ⊂ Q and y ∈ Q′. Let {e_{1}, …, e_{k}} be the standard orthonormal basis of R^{k}. Pick a basis {v_{1}, …, v_{m}} of Q so that v_{j} = e_{j} for all j ≤ k and π(v_{j}) = 0 if j > k. After taking a subsequence, we may find bases {v_{1i}, …, v_{mi}} for the tangent spaces to T′ at (x_{i}, y_{i}) so that π(v_{ji}) = e_{j} for all j ≤ k, π(v_{ji}) = 0 for all j > k and lim_{i→∞} v_{ji} = v_{j}. Let v_{ji} = u_{ji} × w_{ji} for vectors u_{ji} ∈ R^{k} and w_{ji} ∈ R^{n}. likewise, let v_{j} = u_{j} × w_{j}.

We know that 0 = f(x_{i}, y_{i}) = h(x_{i})+ρ(y_{i}). Let α_{i} < 0 be such that h(x_{i})df_{0}|_{T} = α_{i}dr|_{T} at (x_{i}, y_{i}). Then

h(x_{i})∇h · u_{ji} = h(x_{i})df_{0}(v_{ji}) = α_{i}dr(v_{ji}) = 2α_{i}(x_{i} · u_{ji} + y_{i} · w_{ji}).  (∗)

For j > k we thus get 0 = y_{i} · w_{ji} and hence, 0 = y · w_{j} for all j > k. Consequently, y is perpendicular to Q.

Suppose there is a c > 0 so that |y_{i}|/|x_{i}| > c for all i. Since ρ(y_{i}) = −h(x_{i}) and h(x_{i}) → 0 at least as fast as |x_{i}|^{2}, this means that ρ(y) = 0. Thus y ∈ Q. But we have shown that the unit vector y is perpendicular to Q, so we have a contradiction.

So we may assume that lim_{i→∞} |y_{i}|/|x_{i}| = 0. By (∗) we know that

h(x_{i})∇h(x_{i})/(2α_{i}) = x_{i} + W_{i}y_{i}

where W_{i} is the k×n matrix with j-th row w_{ji}. So lim_{i→∞} h(x_{i})∇h(x_{i})/(2α_{i}|x_{i}|) = x since W_{i} → 0 and |y_{i}|/|x_{i}| is bounded. Consequently, lim_{i→∞} ∇h(x_{i})/|∇h(x_{i})| = ±x so by nondepravity we know that h(x_{i}) x_{i} · ∇h(x_{i}) > 0 for all large enough i. But on the other hand,

h(x_{i}) x_{i}·∇h(x_{i}) = 2α_{i}(|x_{i}|^{2}+x_{i}·W_{i}y_{i}) = 2α_{i}|x_{i}|^{2}(1+(x_{i}/|x_{i}|)·W_{i}y_{i}/|x_{i}|) ≈ 2α_{i}|x_{i}|^{2} < 0

which is a contradiction. □

Finally we must prove a lemma about locally continuously extending controlled vector fields.

References

[B] Bekka, Continuous Vector Fields and Thom’s Isotopy Theorem, London Math. Society (1991).

[GM] M. Goresky and R. MacPherson, Stratified Morse Theory, vol. 14, Springer-Verlag, Ergebnisse Math., 1988.

[K1] H. King, Thesis, Univ. of Ca., Berkeley (1974).

[K2] H. King, Topological Type of Isolated Critical Points,, Annals of Math. 107 (1978), 385– 397.

[K3] H. King, Topological Type in Families of Germs, Inventiones Math. 62 (1980), 1–13.

[K4] H. King, Topological Invariance of Intersection Homology without Sheaves, Topology and it Applications 20 (1985), 140–160.

[Sh] Shiota, Piecewise Linearization of Real Analytic Functions, Publ. R.I.M.S. 20 (1984), 724–792.

[S] L. Siebenmann, Deformation of Homeomorphisms on Stratified Sets, Comm. Math. Helv. 47 (1972), 123–163.

Department of Mathematics, University of Maryland, College Park, MD 20742

   E-mail address: hck@ frances.umd.edu