Cone-decompositions and degenerate critical points ^{∗}

Octavian Cornea Université de Lille 1

U.F.R. de Mathématiques & U.R.A 751 au CNRS 59655 Villeneuve D’Ascq, France

cornea@gat.univ-lille1.fr

June 26, 1997

Abstract

Let M be a compact, smooth, 2-connected, 2m-dimensional manifold with ∂M simply connected. If M has the homotopy type of an m-dimensional CW -complex, then it supports a smooth, self-indexed function, maximal, constant and regular on ∂M with at most cat(M)+2 critical points all of which are of a certain ”reasonable” type. To such a critical point there corresponds, homotopically, the attachment of a cone. Conversely, to a cone attachment we may associate, under certain dimensionality and connectivity conditions, a ”reasonable” critical point.

1 Introduction

What is the least number of critical points of a smooth function on a given, smooth manifold M ? When ∂M ≠ ∅ we consider only those functions that are regular, constant and maximal on ∂M. With this restriction denote this minimal number by crit(M).

∗1991 Mathematics Subject Classification: Primary 55M30, 57R70 Secondary 58E05, 55P50, 55P62

It was early discovered (by Lusternik and Schnirelmann [20]) that an answer to this basic question has to involve some homotopy theoretical elements. Indeed, recall that the Lusternik-Schnirelmann category of a CW - complex X, cat(X), is defined as the least n such that there are n + 1 open sets, contractible in X, whose union covers X, [15,20]. It is a homotopical invariant and cat(M)+1 ≤crit(M).

To get more precise results one would like to estimate the difference crit(M)−cat(M)−1. The main aim of this paper is to solve a ”stable” version of this problem (a form of which was raised by Palais [27, pag. 116]) in the 2-connected case.

We prove that: if X is a finite, 2-connected, m-dimensional CW -complex, then on any compact, 2m-dimensional manifold with simply connected boundary having the homotopy type of X there is a self-indexed function with at most cat(X)+2 critical points.

Returning to the manifold M fixed above, it results that, if it is 2connected, then for any t ≥ dim(M), there is on M × D^{t} a self-indexed function with at most cat(M)+2 critical points.

As our critical points are (in general) degenerate we mention that by a self-indexed function we mean one whose critical points x_{i}, i ≥ −1 can be ordered in such a way that x_{−1} is a minimum, x_{i} is of critical value i and, for i ≥ 0, the hessian of f at x_{i} is of index at least i.

Notice that the statement above is just one short of the best possible. The reason is that, in our construction, the central role is played not by cat(−) but by an approximation of it, the cone-length (also known as the strong L.S.-category) [5,10,19], that we will now recall.

Homotopical arguments take place in the pointed category of spaces with the homotopy type of CW -complexes.

Definition 1 The cone-length of X, Cl(X), is the least n for which there are cofibration sequences:

Z_{i} ⟶ X_{i} ⟶ X_{i+1}

with 0 ≤ i < n, X_{0} ≃ ∗, X_{n} ≃ X.

The connection between the cone-length and the Lusternik-Schnirelmann category is provided by the inequality [33]: cat(X)≤ Cl(X)≤ cat(X)+1.

There are very few known examples of CW -complexes with cat(X)≠Cl(X). Strangely enough all of those examples appear for cat(X)= 1 (co-H-spaces that are not suspensions).

In the second section we establish a direct relationship between conelength and critical points by analyzing what happens homotopically at the passage through a ”reasonable” critical point. More precisely, let (M^{(n+1)}; V_{0}, V_{1}) be a compact, smooth, manifold triad: M is a smooth, compact, (n+1)-dimensional manifold whose boundary is the disjoint union of V_{1} and V_{0}. Consider a smooth function f : M ⟶ R which is regular on the boundary of M, constant and minimal on V_{0}, constant and maximal on V_{1}. We call below this type of function a triad function. The basic question is: how can one describe the homotopy type of M in terms of that of V_{0} and some information on the critical set of f inside M ?

Of course, if f has exactly one critical point inside M which is nondegenerate of index k, by elementary Morse theory, we know that M is homotopy equivalent with the space obtained by attaching to V_{0} a cell of dimension k.

The results of the second section imply that, if f has only one critical point which is ”reasonable”, then there is a homotopy cofibration sequence:

Z ⟶ V_{0} ⟶ M

A critical point P is ”reasonable” if it is isolated and the singular hypersurface f^{−1}(f(P)) is Whitney stratified with two strata f^{−1}(f(P)) − P and P (see the definition in section 2). This happens if f is analytic (with respect to some chart) around P or, more generally, if P is a non depraved critical point [11]. In particular this applies to the Morse case. We will see that the homotopy type of Z is independent of the different choices made in its construction. We call it the A-index of P, A_{f}(P). In fact, we prove a more general result concerning ”reasonable” critical manifolds (for the Morse case compare with [1]). In that case, the cofibration sequence is replaced by a certain push-out. Moreover, this homotopy push-out is derived from one defined in the differentiable category.

Incidentally, this implies a form of the L.S.-inequality: a smooth function on a smooth compact manifold N which is constant, maximal and regular on the boundary and has only ”reasonable” critical points cannot have less than Cl(N)+1 of them.

The next point is to see whether in the presence of a cofibration sequence as above one might construct a function with exactly one critical point in the interior of M. Moreover, if this is possible, could one assume that the critical point is reasonable of A-index the homotopy type of Z ?

In the third section we construct the promised functions with few critical points by first showing that: if V_{0}, V_{1}, Z are simply connected, finite complexes and 2dim(Z) ≤ dim(M) − 2, then the answers at the questions above are all affirmative.

It is shown in [5] that, in the definition of the cone-length, one may assume that the space Z_{i} is an i-th order suspension. In our context, this translates into the fact that the functions constructed are self-indexed.

There are two different flavors present in this paper: one consists of exploiting some homotopy-theoretic properties of the cone-length along the lines of [3,4,5]; the other, a more analytic ingredient, has at its center the techniques developed by Takens in [34] (in particular, his method to construct critical points in the presence of a ”nice filling”).

We also make use of techniques of Conley index theory [2,30]. Indeed, the relation with this theory is apparent once one notes that the homotopy type of the space M/V_{0} is precisely the Conley index of the maximal invariant set with respect to the flow induced by −∇f (for some riemannian metric on M). Hence, when the critical point P is ”reasonable” this Conley index is ΣA_{f}(P). Moreover, we will see that the A-index verifies certain formulas implying corresponding known formulas for the Conley index.

The first result of the second section shows how to obtain some informations concerning the Conley index even in the non-”reasonable” case.

Acknowledgements. I thank John Harper who first mentioned the Conley index theory to me, Chris McCord for some useful hints, Steve Halperin and Joe Neisendorfer for stimulating discussions and continuous encouragement, E.N. Dancer for pointing out to me [29] and Howard Marcum who, after the publication of [5], pointed out to me [21] that has some overlaps with [5] and is relevant for a good understanding of the cone-length.

2 Passing through a critical point.

In the first place we analyze the case when the singular set of f is general.

We start by recalling a few elements of Conley index theory [2,30] for the case of a continuous flow γ : N ×R ⟶ N , N being a compact metric space. Let S ⊂ N be an isolated invariant set. A pair (A, B) of compact sets in N is an index pair for S in N if B ⊂ A, A − B is a neighbourhood of S, S is the maximal invariant set in the closure of A − B, B is positively invariant in A and, if for x ∈ A there is some t ≥ 0 such that γ_{t}(x) ∉ A, then there exists a τ > 0 with γ_{t}(x) ∈ A for 0 ≤ t ≤ τ and γ_{τ}(x) ∈ B, [30].

The basic result in this theory is the existence of index pairs inside any isolating neighbourhood and the fact that the homotopy type of the pointed space A/B is independent of the choices involved. This homotopy type is called the Conley index, c_{γ}(S) of S.

From now on let (M^{(n+1)}; V_{0}, V_{1}) be a smooth, riemannian, compact triad and let f : M ⟶ R be smooth, constant and minimal on V_{0}, constant and maximal on V_{1}, regular on the boundary of M. For a collared neighbourhood, U, of ∂M let S_{U} be the maximal invariant set in M − Int(U) with respect to the flow induced by −∇f . The regularity of f on the boundary implies that when U becomes small enough S_{U} stabilizes. We denote by S the resulting invariant set; it is closed and disjoint from the boundary.

All the facts concerning the Conley index can be easily adapted to this context. We denote the index of S with respect to the flow γ induced by −∇f by c_{f}(S). Let M^{−}(S) be the set of the points, x, of M that verify lim_{t→−∞} γ_{t}(x) ∈ S; similarly, M^{+}(S) is the set of those points x that have the property: lim_{t→∞} γ_{t}(x) ∈ S.

Proposition 2.1 If, in the setting above, T is a closed neighbourhood of M^{−}(S) then M is, up to homotopy, a retract of V_{0} ⋃ T .

Proof. Notice that the pair (M, V_{0}) is an index pair for S with respect to γ and the pair (M, V_{1}) is an index pair for S with respect to the inverse flow, γ′, (the flow induced by ∇f). By [30, Lemma 5.3] there is a continuous function: g : M ⟶ [0, 1] such that g^{−1}(0) = V_{1}, g^{−1}(1) = M^{−}(S) and such that t > 0, 0 < g(x) < 1, γ′_{τ}(x) ∈ M − V_{0} for τ ≤ t, t ≠ 0, implies g(γ′_{t}(x)) < g(x). Let L_{ϵ} = g^{−1}([ϵ, 1]), 0 < ϵ < 1. This is again a neighbourhood of M^{−}(S) and we take ϵ small enough such that it is included in T

Let H_{ϵ,δ} = L_{ϵ} ⋃(V_{0})_{δ} where (V_{0})_{δ} = f^{−1}([f(V_{0}), f(V_{0}) + δ]). We choose δ small enough for the intersection (V_{0})_{δ} ⋂ S to be void. Define s_{δ} : M ⟶ R by s_{δ}(x) = min{t : γ_{t}(x) ∈ H_{ϵ,δ}}. This function is continuous. Indeed, let u(x) = min{t : γ_{t}(x) ∈ L_{ϵ}} and let v(x) = min{t : γ_{t}(x) ∈ (V_{0})_{δ}}. Both these two functions are continuous in the interior of their domains. This shows that the only continuity problems for s_{δ} might appear at points where γ_{s_{δ}(x)}(x) belongs to the intersection of (V_{0})_{δ} and L_{ϵ}. However, it is easy to see that there is a neighbourhood of such a point where both u and v are defined. Hence s_{δ} is continuous there also.

Now, it is easy to see that s_{1/n} is uniformly convergent to s = s_{0}. Hence, this last function is continuous too. Let J = max{s(x) : x ∈ M} and let k : M × [0, J] ⟶ M, be defined by k(x, τ) = γ_{τ}(x) for τ < s(x) and k(x, τ) = γ_{s(x)}(x) in the other case. As points in H_{ϵ,0} are not moved by this deformation of M and as the continuity of s implies that of k we see that H_{ϵ,0} is a deformation retract of M. In particular, M is, up to homotopy, a retract of V_{0} ⋃ T.

Corollary 2.2 In the setting above, suppose that for some δ the intersection of (V_{0})_{δ} = f^{−1}([f(V_{0}), f(V_{0}) + δ]) with S is void and that the intersection of T and (V_{0})_{δ} is an NDR in T, then c_{f}(S) is a retract of T/((V_{0})_{δ} ⋂ T).

In most cases it is difficult to impose topological restrictions on the index pairs that define the Conley index. It is much easier to do that for neighbourhoods T as above.

Corollary 2.3 If S is a connected, closed, critical submanifold, then cat(c_{f}(S))≤cat(S)+1. In particular, if S is a critical point then its Conley index has the homotopy type of a co-H-space.

Proof. Indeed, we can take T to be a union of a tubular neighbourhood L of S and (V_{0})_{f(S)−δ}. For δ small enough this contains M^{−}(S) because this set is compact and its intersection with the singular hypersurface f^{−1}(f(S)) reduces to S. We can also assume that the intersection between the boundaries of the tubular neighbourhood used and that of (V_{0})_{f(S)−δ} is transverse. This implies that c_{f}(S) is a homotopy retract of L/(L ⋂(V_{0})_{f(S)−δ}). Now L has the homotopy type of S and the Lusternik-Schnirelmann category does not increase with respect to retracts.

Remark. The result concerning an isolated critical point follows also from results of Dancer [6,7,8]. It was independently obtained by Jonathan Pears in his thesis [28].

We specialize below to the case of a certain type of critical manifold. In that case we will construct some more efficient neighbourhoods of the critical set.

Definition 2 A critical submanifold S of M relative to f : M ⟶ R is reasonable if it is connected and (f^{−1}(f(S))−S, S) is a Whitney stratification of f^{−1}(f(S)).

Remarks. a. We recall that for (f^{−1}(f(S)) − S, S) to be a Whitney stratification for f^{−1}(f(S)) means that f^{−1}(f(S)) − S is a (n-dimensional) submanifold of M and for any sequences of points: p_{i} ∈ f^{−1}(f(S)) − S and q_{i} ∈ S such that they both converge to p ∈ S with the secant lines p_{i}q_{i} converging to l and the tangent spaces T_{p_{i}}(f^{−1}(f(S))) converging to τ we have l ⊂ τ [37].

b. If S is an isolated critical point of a locally analytic function, then it is reasonable. The same applies to a non-depraved critical point [11].

c. If g : N ⟶ M is a locally trivial fibration and S is reasonable for f : M ⟶ R, then g^{−1}(S) is reasonable for f ∘ g.

Assume the context of the above definition. In particular, S is reasonable. Let H be the singular hypersurface f^{−1}(f(S)).

Let U be a tubular neighbourhood of S such that there is a system of local charts for S, V_{i}, and a riemannian metric on M with the property that locally U|_{V_{i}} = {(x, y) : ∥y∥ ≤ 1}, V_{i} = {(x, y) : y = 0}. For 0 ≤ ϵ ≤ 1 we denote by U_{ϵ} the tubular neighbourhood of S which, with respect to the same local charts, is described by U_{ϵ}|_{V_{i}} = {(x, y) : ∥y∥ ≤ ϵ} (hence U_{1} = U, U_{0} = S). Let p_{ϵ} be the projection in U_{ϵ}. For our purposes the following facts are of great significance:

Lemma 1. For ϵ small enough ∂U_{τ} intersects H transversely for all 0 < τ ≤ ϵ.

We fix such an ϵ > 0.

Lemma 2. The pair (U_{ϵ} , U_{ϵ} ⋂ H) is homeomorphic with the mapping cylinder (∂U_{ϵ}×[0, 1] ⋃_{p_{ϵ}} S , (∂U_{ϵ} ⋂ H)×[0, 1] ⋃_{p_{ϵ}} S) and this homeomorphism identifies ∂U_{ϵ} × [ϵ, τ] with U_{ϵ} − Int(U_{τ}).

Remarks. a. These two results are classical for the case of an isolated critical point of an analytic function [25].

b. In the present context the first fact is classical and immediate from the reasonability condition. The second one follows, for example, from the same type of arguments as those used in [25, Theorem 2.10] for the isolated critical point case. Indeed, the situation described there is exactly what happens here in the normal direction.

c. It follows from the basic properties of a Whitney stratification that the normal projection in U_{ϵ} induces a submersion of ∂U_{ϵ} ⋂ H onto S.

Given two neighbourhoods U, V, of the type discussed above let ϵ > 0 be as in Lemma 1 with respect to both U and V.

Lemma 3. The two manifold pairs (∂U_{ϵ} ⋂ f^{−1}(−∞, f(S)], ∂U_{ϵ} ⋂ H) and (∂V_{ϵ} ⋂ f^{−1}(−∞, f(S)], ∂V_{ϵ} ⋂ H) are invertibly cobordant.

This is again the obvious extension of the case of an isolated critical point [16,17].

We recall that two manifolds A, B are invertibly cobordant if there are cobordisms W and W′ between A and B such that pasting W to W′ following the A end we get, up to homeomorphism, the cylinder B × [0, 1] and pasting W to W′ following the B end we get the cylinder A × [0, 1] [31,32]. It is easy to see that if A and B are invertibly cobordant then they are of the same homotopy type (indeed, A × R is homeomorphic to B × R which follows by looking at the telescope:.....W ⋃_{A} W′ ⋃_{B} W ⋃_{A} .... , [16]).

Lemma 3 follows from Lemma 2 by remarking that inside U_{ϵ} we can find V_{η} for η small enough and inside V_{η} we can embed U_{τ} for τ small enough.

Before stating the main theorem of this section we need a definition.

Definition 3 Consider compact, smooth manifolds A^{n}, B^{n}, D^{k}, D^{′n+1} with ∂B = ∂D = ∅ ≠ ∂A, k < n and smooth embeddings a : A ⟶ ∂D′, j : A ⟶ B, d : D ⟶ D′ with d a homotopy equivalence. Form C by pasting B × [0, 1] to D′ along A using j × {1} and a. We call the diagram:

[figure: A; a; B; j; D; D′; C; d]

a smooth left push out.

Remark. a. Obviously, a smooth left push out induces a homotopy push out:

[figure: A; a′; B; j; D; C]

The map a′ is given by the composition of a with the inverse of the homotopy equivalence d.

b. The smooth structure on C above is provided by rounding the corners of the topological manifold obtained by pasting. The fact that we get a unique structure in this way is ensured by [10].

We return to the setting of Proposition 2.1: we are given the triad (M^{(n+1)}; V_{0}, V_{1}), the function f : M ⟶ R and the invariant set S.

Theorem 2.4 Suppose that S is a reasonable, simply connected, critical manifold and let U_{ϵ} be a tubular neighbourhood of S as in Lemma 1.

There is a smooth left push out inducing the homotopy push out square:

[figure: Z; V_{0}; S; M]

with Z invertibly cobordant to ∂U_{ϵ} ⋂ f^{−1}(−∞, f(S)].

Proof. The idea is to construct a special type of neighbourhood of S, G, with the following properties:

a. G is a smooth manifold except some corners on the boundary.

b. If Z = V_{0} ⋂ G, then Z is a smooth, n-dimensional manifold with nonvoid boundary embedded in V_{0} and in ∂G in such a way that ∂G has corners along ∂Z.

c. S is homotopy equivalent to G.

d. V_{0} ⋃ G is, up to rounding the corners, diffeomorphic to M.

e. Z is invertibly cobordant to ∂U_{ϵ} ⋂ f^{−1}(−∞, f(S)].

f. G has the same properties relative to the function −f and the triad (M; V_{1}, V_{0})

The existence of G with all of the above properties implies the theorem. In fact, f is not necessary for the theorem but will be used for some of its corollaries.

Let now ϵ be as in Lemma 1. For τ ≤ ϵ we denote H_{τ} = H ⋂ U_{τ}, T_{τ} = H ⋂ ∂U_{τ}. Let G_{τ} consist of the points x ∈ M for which either there is a t with the poperty γ_{t}(x) ∈ H_{τ} or lim_{t→∞}(γ_{t}(x)) ∈ S or lim_{t→−∞}(γ_{t}(x)) ∈ S}. This is a type of ”cylindrical” neighbourhood used in [29], [34] and [7,8,9] in the case of an isolated critical point. The neighbourhood G that we are looking for is of this type. Let K_{τ} = {x ∈ M : ∃t, γ_{t}(x) ∈ T_{τ}}. Notice that as T_{τ} separates H and as f has no critical points in M besides the points of S we see that K_{τ} is a smooth hypersurface of M, transverse to ∂M, diffeomorphic to T_{τ} × [0, 1] and separating M − ∂M into two parts (not connected in general) one of which is Int(G_{τ}). Denote by T′_{τ} the intersection of K_{τ} with V_{0}. This also separates, and one side of it is Z_{τ} = G_{τ} ⋂ V_{0}. Clearly, Z_{τ} is a smooth manifold with boundary and G_{τ} is an n+1 dimensional manifold whose boundary has corners. So for G_{τ} we have verified properties a and b above.

We will now verify the property c. From Lemma 2 it follows that S is a deformation retract of H_{τ}. All we have to check is that H_{τ} and G_{τ} have the same homotopy type. Consider the diagram:

[figure: S; H_{τ}; H_{τ}/S; S; G_{τ}; G_{τ}/S]

The horizontal sequences are cofibration sequences and the vertical arrows inclusions. Recall that S is simply connected. It follows that H_{τ} also is. Hence, to deduce that the middle arrow is a homotopy equivalence, it is enough to see that the inclusion H_{τ}/S ↪ G_{τ}/S is one (all spaces here admit CW -structures). We will see that, in fact, H_{τ}/S is a deformation retract of G_{τ}/S the deformation being induced by γ. We construct this deformation in two steps. Write G_{τ} = G′ ⋃ G′′ where G′ = G_{τ} ⋂ f^{−1}[f(S), +∞), G′′ = G_{τ} ⋂ f^{−1}(−∞, f(S)].

The flow γ induces a deformation of G′, γ′ : G′ × [0, ∞) ⟶ G′ defined by γ′(x, t) = γ_{t}(x) if f(γ_{t}(x)) > f(S) and γ′(x, t) = γ_{T (x)}(x) in the opposite case; here T (x) is defined by f(γ_{T (x)}(x)) = f(S). By precomposing with a diffeomorphism [0, 1) ⟶ [0, ∞) we may take γ′ : G′ × [0, 1) ⟶ G′. It is clear that γ′ is continuous, that it does not move H_{τ} and that all the points in G′ − M^{+}(S) arrive in H_{τ} in time < 1. However, there is no reason why γ′ should extend to a deformation retract of G′ to H_{τ}. On the other hand consider γ′_{1} : G′_{τ}/S ×[0, 1] ⟶ G′_{τ}/S defined by γ′_{1}(x, t) = γ′(x, t) when t ≠ 1, γ′_{1}(x, 1) = lim_{t→1}γ′(x, t) when x ∉ M^{+}(S) and γ′_{1}(x, 1) = ∗ if x ∈ M^{+}(S). It is easy to see that this is continuous, γ′_{1}(x, t) = x for all t and x ∈ H_{τ}/S, and γ′_{1}(x, 0) = x for all x ∈ G′_{tau}/S.

Obviously, an analogous construction can be made with respect to G′′. We pass to the proof of d. We assume now that τ < ϵ. Notice that

the set G_{ϵ,τ} = G_{ϵ} − Int(G_{τ}) is homeomorphic to K_{ϵ} × [0, 1] and hence to T_{ϵ} × [0, 1] × [0, 1]. Let D_{ϵ,τ} = T_{ϵ} × {(x, y) ∈ [0, 1] × [0, 1] : x ≤ y}. We may assume that the parametrizations used are such that x is a coordinate in K_{ϵ}, y a coordinate in H and ∇f is transverse to T_{ϵ} × (t, t). Note that, up to rounding the corners, F_{ϵ,τ} = V_{0} ⋃ D_{ϵ,τ} ⋃ G_{τ} and V_{0} ⋃ G_{τ} are diffeomorphic. Remark now that F_{ϵ,τ} is diffeomorphic (up to rounding the corners) to M. Indeed, this follows from the fact that the flow γ restricted to M − Int(F_{ϵ,τ}) has no invariant points.

To establish property e we begin by defining J_{τ} = G_{τ} ⋂ f^{−1}[f(S) − τ, f(S)+τ]. It is easy to see that J_{ϵ}−Int(J_{τ}) is homeomorphic to ∂G_{ϵ}×[0, 1]. Moreover this homeomorphism restricts to a homeomorphism of ∂J_{ϵ} ⋂ f^{−1}(−∞, f(S)] with B_{ϵ} = (Z_{ϵ} ⋃(K_{ϵ} ⋂ f^{−1}(−∞, f(S)])). Inside any neighbourhood U_{ϵ} as in Lemma 1 we can embed a J_{τ} for τ small enough. Indeed, if this were not the case there would exist sequences x_{n} ∈ T_{1/n} and t_{n} ∈ R with | f(γ_{t_{n}}(x_{n})) − f(S) |≤ 1/n and γ_{t_{n}}(x_{n}) ∉ U_{ϵ}. By passing to subsequences we may assume that x_{n} and γ_{t_{n}}(x_{n}) both converge respectively to x ∈ S and y ∈ H − Int(U_{ϵ}). This leads to a contradiction because it implies that y is invariant. Inside J_{τ} we can embed a set U_{ϵ′} for some small ϵ′. It follows that B_{ϵ} and ∂U_{ϵ} ⋂ f^{−1}(−∞, f(S)] are invertibly cobordant. The claim follows by remarking that K_{ϵ} ⋂ f^{−1}(−∞, f(S)] is diffeomorphic to T_{ϵ} × [0, 1] and T_{ϵ} = ∂Z_{ϵ}.

From these constructions it follows that any G_{τ} satisfies a,b,c,d,e and the same properties with respect to −f. Hence, condition f is automatically satified.

We will call the homotopy type of Z in the theorem above the A-index of S, A_{f}(S). We extend this definition to the case when Z = ∅ (which is equivalent to V_{0} = ∅) by taking A_{f}(∅) = ∗.

Remarks. a. If S is a point parts of the theorem remain valid in the three dimensional case for general functions [34]. For homogenous functions it is much easier to construct G with the properties described in the above proof [2]. Also, as suggested by E.N.Dancer, the hypothesis H of Rothe [29] (which is weaker than the reasonability condition but more dependent on the metric), is sufficient for parts of the theorem.

b. The A-index of a reasonable critical point P of f depends only on the right-equivalence class of the singularity of f at P.

c. It is likely that ”controlled” (in the sense of control data [22]) adaptations of this theorem and of its proof remain true for a critical set S which is Whitney stratified and also in the case of a stratified mapping (by analogy with the ”stratified Morse theory” of Goresky and MacPherson [11]).

d. The non-reasonable case has been recently analyzed by Thomas Morgenstern in his thesis [26]. His work is relevant also for the Corollary 2.7 that follows.

Corollary 2.5 Suppose that the compact manifold M is connected and let f : M ⟶ R be a smooth function that has m reasonable critical points in the interior of M and is constant, maximal and regular on the boundary of M, then m ≥Cl(M)+1.

Proof. Clearly, we may assume that all the local minimum points correspond to the same critical value and that all the other critical points are unique on their critical levels. By applying the theorem for each critical point (except the minima) in turn, we get homotopy cofibration sequences: Z_{i} ⟶ X_{i} ⟶ X_{i+1} with 0 ≤ i < m, X_{0} is a disjoint union of points (of number equal to that of the minima) and X_{m} ≃ M. These cofibration sequences differ from those used in the definition of the cone-length by the fact that X_{0} (and in consequence, some of the X_{i}’s) may not be connected. However, because X_{m} is connected we can modify each such cofibration sequence in such a way that each of the X_{i} becomes connected and such that the number of cofibrations and X_{m} do not change. Indeed, it is enough to use the following remark. Consider a cofibration sequence of not necessarily connected spaces: A ⟶ B ⟶ C. There is another cofibration sequence A′ ⟶ B′ ⟶ C′ with B′ (and C′) obtained from B (respectively C) by collapsing the respective 0-skeleta to a point.

It is easy to state a relative version of the corollary above. We first extend the definition of the cone-length.

Definition 4 Let f : Y ⟶ X be a map of CW -complexes. The relative cone-length of X with respect to f, Cl(X, Y), is the least n such that there are cofibrations Z_{i} ⟶ X_{i} ⟶ X_{i+1}, X_{0} ≃ Y, X_{n} ≃ X and the composite Y ⟶ X_{i} ↪ X is homotopic with f for all 0 ≤ i ≤ n.

Clearly, Cl(X)=Cl(X, ∗).

Assume that (W^{(n+1)}; V_{0}, V_{1}) is a manifold triad such that V_{0} ≠ ∅, then any triad function f : W ⟶ R which has only reasonable critical points has at least Cl(W, V_{0}) of them.

Notice that, in the cone-decompositions obtained above, the spaces Z_{i} have the homotopy type of finite complexes.

The following consequences of the theorem are immediate:

Corollary 2.6 In the setting of the theorem: a. There is a homotopy cofibration sequence:

Z ⟶ S ⟶ c_{f}(S)

In particular, if S is a point c_{f}(S) = ΣA_{f}(S). b. A_{f}(S) ⋃_{T_{ϵ}} A_{−f}(S) ≃ ∂U_{ϵ}. c. If g : N ⟶ M is a locally trivial fibration having a simply connected fibre, F, then there is a fibration F ⟶ A_{f∘g}(g^{−1}(S)) ⟶ A_{f}(S). This fibration is trivial when S is a point and c_{f∘g}(g^{−1}(S)) = (F ∧ c_{f}(S)) ∨ c_{f}(S)

Proof. The first two points are obvious as well as the first part of the third. For the second part remark that by applying the theorem to the triad N and the function f ∘ g we obtain a cofibration sequence:

A_{f∘g} = A_{f} × F ⟶ F ⟶ c_{f∘g}

By collapsing F in the first two terms the cofibre does not change (up to homotopy) and this gives the result.

Suppose f : M ⟶ R, g : N ⟶ R are two smooth functions. We assume that S is a simply connected reasonable critical manifold of f and S′ one of g. Let f + g : M × N ⟶ R be defined by (f + g)(x, y) = f(x) + g(y). Assume that S × S′ is a reasonable critical manifold for f + g. Of course M ×N is not a triad in the above sense, however the A-index is a local notion so it is well defined for S × S′. To ease notation we will write: A_{f} = A_{f}(S), A_{g} = A_{g}(S′) and A_{f+g}(S × S′) = A_{f+g}.

Corollary 2.7 There is a homotopy push out square:

[figure: A_{f} × A_{g}; S × A_{g}; A_{f} × S′; A_{f+g}]

In particular, if S and S′ are points, we have A_{f+g} = A_{f} ∗ A_{g}.

Proof. Suppose, to simplify notation, that f(S) = g(S′) = 0. Let J_{τ}, J′_{τ} be neighbourhoods of S and, respectively, S′ defined as in the proof of the theorem. If U_{ϵ} is a neighbourhood of S × S′ as in Lemma 1, for τ small enough we can embed inside U_{ϵ} a product neighbourhood of the type J_{τ} × J′_{τ}. Let P_{τ} = {(x, y) ∈ ∂(J_{τ} × J′_{τ}) : f(x) + g(y) ≤ 0}. Notice that P_{τ} is a piecewise smooth manifold. Indeed, projecting the gradient of f + g on each of the smooth pieces of ∂(J_{τ} × J′_{τ}) we see that each such piece which cuts the singular hypersurface Y = (f +g)^{−1}(0) does so transversely (we take the product metric on M × N). It follows, by an argument similar to that in the proof of the theorem, that P_{τ} is invertibly cobordant to {z ∈ ∂U_{ϵ} : (f +g)(z) ≤ 0}. Recall that ∂J_{τ} is homeomorphic to the union: W_{τ} ⋃ Q_{τ} ⋃ R_{τ} where W_{τ} = J_{τ} ⋂ f^{−1}(−τ), Q_{τ} = J_{τ} ⋂ f^{−1}(+τ) and R_{τ} is homeomorphic to a cylinder over ∂W_{τ} = ∂Q_{τ}; W_{τ} has the homotopy type of the A-index of S. A similar decomposition is valid in the case of the invariant set S′. Notice that P_{τ} ⊂ ((W_{τ} ⋃ R_{τ}) × J′_{τ}) ⋃(J_{τ} × (W′_{τ} ⋃ R′_{τ})) = L. Consider the set K = (R_{τ} ×J′_{τ}) ⋃(J_{τ} ×R′_{τ}). Denote γ′ the flow induced by −∇(f+g)(x, y). Let b : K ⟶ R, b(x, y) = min{t ∈ R : γ′_{t}(x, y) ∈ (W_{τ} ×J′_{τ}) ⋃(J_{τ} ×W′_{τ})}. As the vector −∇(f + g)(x, y) belongs to the tangent space to K when (x, y) ∈ K it is easy to see that this function is continuous and with its help we can define a flow-induced deformation retract of L onto (W_{τ} × J′_{τ}) ⋃(J_{τ} × W′_{τ}): we move points in K following γ′ and once we reach (W_{τ} ×J′_{τ}) ⋃(J_{τ} ×W′_{τ}) we do not move, points in L − K do not move at all. This deformation restricts to one of P_{τ}. This shows that P_{τ} ≃ (W_{τ} ×J′_{τ}) ⋃(J_{τ} ×W′_{τ}). We now note that W_{τ} × J′_{τ} ⋂ J_{τ} × W′_{τ} = W_{τ} × W′_{τ}. Recalling that J_{τ} and J′_{τ} have respectively the homotopy types of S and S′ we obtain the desired push out.

Remark. When S and S′ are points and one of them is non-degenerate we recover a formula of King [18].

3 Attaching cones

The purpose of this section is to construct the functions with few critical points promised in the introduction.

To begin, we recall the technique introduced by Takens [34] to construct degenerated isolated critical points. The reason why we do this explicitly here is two-fold: we want to remark that his technique can be used to obtain critical-manifolds and even reasonable ones.

Definition 5 The square in Definition 3 is a strong, smooth, left push out if, in that context, D′ is the total space of a closed disk bundle over D, d is the inclusion of the zero section and the projection in D′ induces a submersion from a(∂A) onto D. It is a simple, strong, smooth, left push out if, in addition, both A and ∂A are simply connected.

Remarks. 1. By specializing the definition of a strong, left push out we define a strong, smooth, left cofibration sequence to be a sequence A ⟶ B ⟶ C such that C is diffeomorphic to B × [0, 1] ⋃_{A} D^{t} where D^{t} is a disk of dimension t = dim(C). Hence, this gives a nice filling for C in the sense of [34]. We prefer here the language of push outs and cofibration sequences to that of fillings because it better suits our homotopy theoretic approach.

2. The proof of Theorem 2.4 applied to the case of an isolated critical point produces a strong, smooth, left cofibration.

Proposition 3.1 Given a strong, smooth, left push out:

[figure: A; B; D; D′; C]

with D connected, there is a smooth triad function f : C ⟶ R having D as reasonable critical manifold, without critical points in the exterior of D and which is constant and minimal on B × {0} and such that A_{f}(D) ≃ A.

Proof. Denote B_{0} = B×{0} and B_{1} = (B×{1}−Int(A)) ⋃_{∂A}(∂D′−Int(A)). Note that, we may assume that, up to rounding the corners, C = (B_{0} × [0, 1] ⋃ D′) ⋃ B_{1} × [0, 1] the first union being over A. Denote B′_{1} = B_{1} × {1}. By Definition 5, D′ is the total space of a disk bundle over D. Hence there is a radial deformation of D′ to D. This is a homotopy r : D′ × [0, 1] ⟶ D′, that verifies r(x, t) = x for all t and x ∈ D, r(x, 1) ∈ D for all x ∈ D′, r(x, 0) = x for all x ∈ D′ and, for all t ∈ [0, 1], r(x, t) belongs to the fiber through x .

Let H = (B × {1} − Int(A)) ⋃_{∂A} r(∂A, [0, 1]). After rounding the corners of H around ∂A, H becomes a smooth hypersurface except on D. The pair (H − D, D) is a Whitney stratification for H. Indeed, this follows by remarking that any secant vector joining a point in H − D to one in D can be written as the sum of a radial vector and one joining two points in D and using the fact that the projection p : D′ ⟶ D, restricted to ∂A is a submersion to D (this is a particular case of an argument of Goresky [12]).

We can now start to apply the construction of Takens [34, Proposition 2.6]. The first step provides a function f′ which is minimal and constant on B_{0}, maximal and constant on B′_{1}, vanishes only on H ⋃ D′ and is smooth outside D′ and has no critical points there. Using a somewhat bigger tubular neighbourhood of D than D′, D′′, we see that r extends to a map: u : C ⟶ C which restricts to a diffeomorphism of C−D′ onto C−D and is the identity near the boundary of C . We may assume that u is radial in the sense that p(u(x)) = p(x) (in particular, this means that u respects H). Also, we may assume that on D′′ the function f′ is normal to D in the sense that the projection of its gradient on the tangent space of D is null. By precomposing f′ with the inverse of u and extending by continuity the resulting map over D, we get a new function f′′ which has the same properties as f′ relative to the boundary of C, which vanishes only on H, is smooth outside D and has no critical points there.

The next step is to ”smooth” f′′ over D. This is done by applying a parametrized version of Theorem 2.7 of [34]: the space of parameters is D. The existence of such a version is ensured by the fact that, by construction, in a neighbourhood of D, the gradient of f′′ is normal to D. Notice also that by the smoothing procedure of Takens the 0-level hypersurface is not modified.

Remark. It is likely that this result extends to certain cases when D is a Whitney stratified set.

Corollary 3.2 Any homotopy type of a finite complex appears as an A-index of an isolated reasonable critical point. In particular, any finite suspension can be realized as the Conley index of an isolated critical point.

Proof. It is enough to recall that we can represent any such homotopy type by an n-manifold with boundary, embedded in the n-dimensional sphere for n large enough.

Remark. The result concerning the Conley index has been independently obtained by Jonathan Pears in his thesis [28].

We need a second type of construction which shows that, whenever some obvious topological restrictions are satisfied, one can assume that, in a neighbourhood of D, the function constructed in the above proposition has a predefined form.

Lemma 3.3 Assume the setting of Proposition 3.1 . Assume also that the push out is simple and that D and B are simply connected. Moreover, suppose that there is a smooth function g : D′ ⟶ R such that :

1. D is a reasonable critical manifold of g and, with respect to g, D′ is a neighbourhood of D of the type appearing in Lemma 1. Using the notations there, we put U_{1} = D′.

2. g^{−1}(0) ⋂ ∂D′ = ∂A, g^{−1}((−∞, 0]) = A.

Then, there is a τ ≤ 1 such that g|_{U_{τ}} extends to a function f : C ⟶ R with the properties in Proposition 3.1.

Proof. We assume that g(D) = 0. We maintain the notations used in the proof of Proposition 3.1 with the exception that here H is obtained from (B × {1} − Int(A)) ⋃_{∂A} g^{−1}(0) by rounding the corners (along ∂A). Fix τ < 1 and take V, a small tubular neighbourhood of H − U_{τ}. Let V_{ϵ} = g^{−1}[−ϵ, ϵ]−U_{τ}. Take ϵ small enough such that V_{ϵ} ⊂ Int(V). Of course, V_{ϵ} is a bicollared neighbourhood of (H ⋂ U_{1})−U_{τ}. All tubular neighbourhoods here are trivial. Hence we may extend V_{ϵ}, inside V, to a tubular neighbourhood V′ of H − U_{τ} and also g|_{V_{ϵ}} to a function g′ : V′ ⟶ R which takes the value +ϵ on the ”top” part of the boundary of V′ , takes the value −ϵ on the other part of the boundary, and has no critical points in V′. By pasting g and g′ (which agree on V_{ϵ}) we obtain a function g′′ : T = V′ ⋃ U_{τ} ⟶ R. After smoothing the corners of T we see that g′′ is smooth, and its only critical points lie on D. Moreover, C − T is diffeomorphic to the disjoint union of B_{0} × [0, 1] and B′_{1} × [0, 1]. If the gradient of g′′ pointed outside T on the top part of its boundary and inside on the bottom part it is easy to see that we could extend g′′ to a function on C satisfying the required properties. As this might not be the case let J_{γ} be a cylindrical neighbourhood of D of the type described in the proof of Theorem 2.4 and such that J_{γ} ⊂ T. We can construct a neighbourhood L of H inside T, exactly as in the construction of T except that in the place of U_{τ} we use J_{γ}. Moreover a small modification of the ”vertical walls” of J_{γ} (as in the proof of the point d in the proof of Theorem 2.4) suffices to ensure that the gradient of g′′ restricted to the top and bottom parts of the boundary of L has the requiered properties. The simple connectivity conditions are needed to ensure that D and J_{γ} have the same homotopy type and to apply the h-cobordism theorem on each of the two cobordisms whose disjoint union is C − Int(L).

The key step in the construction of functions with few critical points is the following partial converse to Theorem 2.4.

Theorem 3.4 Let (W^{(n+1)}; A_{0}, A_{1}) be a manifold triad with A_{0}, A_{1} simply connected. If there is a homotopy cofibration sequence B ⟶ A_{0} ⟶ W with B simply connected, finite and 2dim(B) ≤ n − 1, then this cofibration sequence is induced by a simple, strong, smooth, left one. In particular, there is a triad function f : W ⟶ R which has a single critical point , P, inside W which is reasonable and A_{f}(P) ≃ B.

Moreover, given a function g : D^{n+1} ⟶ R having at 0 ∈ D^{n+1} a reasonable critical point such that A_{g}(0) = B, we may assume that, on some neighbourhood of P, f and g agree up to a local change of coordinates.

The first step is to point out some sufficient conditions for fusing critical points. The result below will also provide an induction step for the theorem.

Let (M^{(n+1)}; V_{0}, V_{1}) and (N^{(n+1)}; V_{1}, V_{2}) be manifold triads. Let (J; V_{0}, V_{2}) be the composed triad (obtained by pasting M and N along V_{1}).

Proposition 3.5 Assume that V_{0}, V_{1} and V_{2} are simply connected and that there is a commutative diagram:

[figure: ⋁_{1≤i≤t} S^{m}; E; V_{0}; id; M; F; V_{0}; J]

where both lines and the column are homotopy cofibration sequences of which the top horizontal line is a simple, strong, smooth, left cofibration sequence. If 2m ≤ n − 1, m ≥ 2, and E is homotopy equivalent to a finite CW - complex of dimension q with m < n − q − 1, then there is a simple, strong, smooth, left cofibration sequence F′ ⟶ V_{0} ⟶ J′′ which induces the bottom horizontal line and such that F′ is obtained from E by the attachment of t (m,n-m)-handles.

Proof. We begin with a simple homotopical remark.

Lemma 3.6 There is a commutative diagram:

[figure: ⋁_{1≤i≤t} S^{m−1}; ∨α′_{i}; ∗; ⋁_{1≤i≤t} S^{m}; ∨s_{i}; E; V_{0}; id; M; F; V_{0}; J]

with all lines and columns homotopy cofibration sequences and the bottom two lines the same, up to homotopy, as in the diagram above.

Proof of lemma. By inspecting the diagram in the statement of the theorem we see that, if Q is the cofibre of the map E ⟶ F, then ΣQ ≃ ⋁_{1≤i≤t} S^{m+1}. This and the fact that E is 1-connected imply that Q ≃ ⋁_{1≤i≤t} S^{m}. Let {α_{i}} ⊂ π_{m}(F, E) ≃ H_{m}(F, E) ≃ Z^{t} be a set of free generators. Let α′_{i} = ∂α_{i} where ∂ is the boundary map π_{m}(F, E) ⟶ π_{m−1}(E). The map E ⟶ F factors as E ⟶ F′ ⟶^{w} F where F′ is the cofibre of ∨α′_{i}. Clearly w is a homotopy equivalence. We can construct the top left square of the desired diagram and by taking cofibres of all the maps that it contains we complete the construction.

For further use note that the top left square in the above diagram gives t relative homotopy classes β_{i} ∈ π_{m}(V_{0}, E) with ∂β_{i} = α′_{i} where ∂ : π_{m}(V_{0}, E) ⟶ π_{m−1}(E).

We return to the proof of the proposition. The idea is to realize geometrically the cone attachments contained in the diagram of the lemma.

To begin with we will assume that t = 1 and drop the index 1. By the definition of a simple, smooth, strong, left push out, E is an n-manifold such that: E and ∂E are simply connected and we may identify M, up to smoothing the corners, with (V_{0} × [0, 1]) ⋃_{E} D^{n+1}. The union is taken using embeddings of E in V_{0} × {1} and in S^{n} = ∂D^{n+1}. By Poincaré duality we have: H_{r}(E, ∂E) ≃ H^{n−r}(E). As H^{∗}(E) = 0 for ∗ > q we obtain that the pair (E, ∂E) is (n − q − 1)-connected. Write W = (V_{0} × {1}) − Int(E). As both E and ∂E are simply connected it follows (by van Kampen’s theorem) that W is also simply connected and in particular the pair (W, ∂E) is so. Hence, by homotopy excision, the obvious map j : π_{∗}(W, ∂E) ⟶ π_{∗}(V_{0} × {1}, E) is an epimorphism for ∗ ≤ n − q. Thus there is a class β′ ∈ π_{m}(W, ∂E) such that j(β′) = β. As 2m < n we may assume that β′ is represented by a neat embedding r : (D^{m}, S^{m−1}) ⟶ (W, ∂E). The same excision arguments as above show that if T = S^{n} − Int(E), then the map i : π_{m}(T, ∂E) ⟶ π_{m}(S^{n}, E) is an epimorphism. In particular, it follows that there is a class η ∈ π_{m}(T, ∂E) whose image by the boundary map π_{m}(T, ∂E) ⟶ π_{m−1}(∂E) is the class of r(S^{m−1}). We deduce the existence of a neat embedding u : (D^{m}, S^{m−1}) ⟶ (T, ∂E), representing η, such that the restriction of u to S^{m−1} agrees with that of r. We paste r and u along S^{m−1}. We get a topological embedding e of S^{m} in Y = W ⋃_{∂E} T which is smooth except at the corners along S^{m−1}. Its homotopy class in M is precisely s (from the lemma). We now remark that the class of e in π_{m}(Y) is in the image of π_{m+1}(N, Y) ⟶ π_{m}(Y). Indeed this follows from the following three facts: the class of s is in the image of the boundary π_{m+1}(J, M) ⟶ π_{m}(J); the obvious map π_{m+1}(N, Y) ⟶ π_{m+1}(J, M) is an epimorphism (this follows by homotopy excision from the fact that the pair (N, Y) is m-connected and the pair (M, Y) is simply connected); the map π_{m}(Y) ⟶ π_{m}(M) is an isomorphism (this is so because, by Poincaré duality, H_{∗}(M, Y) ≃ H^{n+1−∗}(M, V_{0}) ≃ H^{n+1−∗}(ΣE), as m < n − q − 1 this shows that the pair (M, Y) is at least (m + 1)-connected). As 2(m + 1) ≤ n + 1, by Whitney’s trick we can find an embedding l : (D^{m+1}, S^{m}) ⟶ (N, Y) whose restriction to S^{m} coincides with e.

We return now to the case when t ≥ 1. We assume that the above construction has been done for each of the classes β_{i}. We use exactly the notations above but we add the corresponding index. In particular, we have embeddings l_{i} : (D^{m+1}, S^{m}) ⟶ (N, Y) such that their restriction to S^{m}, denoted by e_{i}, represents s_{i}. As 2m < n we may assume that the images of the e_{i}’s are disjoint. We may assume also that the images of the l_{i}’s are disjoint. Indeed, first suppose that the intersection of the l_{i}’s is transverse. If p is an intersection point of l_{i} and l_{j} we can find a path in l_{i} joining p to a point p_{i} ∈ S^{n} − W and similarly a path in l_{j}. We then join p_{i} to p_{j} by a path in S^{n} −W. As N is simply connected and m+1 ≥ 3 the loop obtained in this way bounds an embedded 2-disk. We may assume that everything in this construction avoids the other l_{k}’s and W. Finally, push the two (m+1)-disks apart along the 2-disk.

The homotopy type of the space obtained by attaching the disks l_{i}(D^{m+1}) to (V_{0} × [0, 1]) ⋃ D^{n+1} is that of J.

Let now L_{i} be a small tubular neighbourhood of l_{i}(D^{m+1}) such that O_{i} = L_{i} ⋂ Y is a tubular neighbourhood of e_{i}(S^{m}) and, moreover, I_{i} = L_{i} ⋂ V_{0} × {1} is a tubular neighbourhood of r_{i}(D^{m}) and G_{i} = L_{i} ⋂ S^{n} is a tubular neighbourhood of u_{i}(D^{m}). We assume also that the various intersections with ∂E are transversal and that the L_{i}’s are pairwise disjoint. Let J′ = ((V_{0} × [0, 1]) ⋃ D^{n+1}) ⋃_{O_{i}}^{1≤i≤t} L_{i}, F′ = E ⋃^{1≤i≤t} I_{i} and D′ = D^{n+1} ⋃_{G_{i}}^{1≤i≤t} L_{i}. Clearly, J′ has the homotopy type of J, D′ is an (n+1)-dimensional disk, and we can also write J′ as the union: V_{0} × [0, 1] ⋃_{F′} D′. Notice that because m ≥ 2 it follows that F′ and ∂F′ are simply connected. Applying the hcobordism theorem to the cobordism J − Int(J′) we deduce that it is trivial and hence, modulo smoothing the corners of J′, J and J′ are diffeomorphic.

Remark. In view of the first section it is easy to see that the proposition gives sufficient conditions when it is possible to fuse critical points. Indeed, for a triad J as above assume we have a triad function f which is constant on V_{1} and has exactly two critical points: one reasonable inside M (denote it by P) and another, non-degenerate of index m + 1, inside N. By the results of the second section we get a cofibration sequence associated to P A_{f}(P) ⟶ V_{0} ⟶ M and also a second cofibration sequence S^{m} ⟶ M ⟶ J. We assume that the different dimensionality and connectivity conditions in the statement of the proposition are satisfied. Under these circumstances if out of our two cofibration sequences we can build a diagram as that in the proposition, then there is a triad function on J with just one reasonable critical point. Note that this is a homotopy theoretic condition.

Proof of Theorem 3.4. For the first part of the theorem apply the proposition following the skeletal filtration of B. Indeed, let ⋁ S^{i} ⟶ B^{(i)} ⟶ B^{(i+1)} be the cofibrations corresponding to this filtration. Here, 1 ≤ i < dim(B), B^{(1)} = D^{n} ⊂ A_{0}, B^{(g)} ≃ B and dim(B^{(i)}) = i. To each such cofibration sequence we can inductively asssociate a diagram as in the lemma:

[figure: ⋁ S^{i}; ∗; ⋁ S^{i+1}; B^{(i)}; (V_{0})_{i}; W_{i}; B^{(i+1)}; (V_{0})_{i}; W_{i+1}]

Here, (V_{0})_{i} is the top boundary component of W_{i−1}. Notice, that in all these diagrams we have i + 1 ≤ dim(B) ≤ (n − 1)/2 and i + 1 < n − (dim(B^{(i)}) − 1. Hence, at each step the proposition may be applied. The connectivity conditions are easily checked.

For the second part of the theorem we need to verify that the existence of the function g in the statement allows to apply Lemma 3.3. In fact, we have constructed above a thickening of B in A_{0} and a simultaneous thickening in S^{n}. To g we may associate another thickening of B in S^{n} by simply looking at the set g^{−1}((−∞, 0]) ⋂ S_{ϵ} (where S_{ϵ} is a sufficiently small n-sphere centered at 0). If this two thickenings agree the lemma can be applied. But this happens because, by the embedding theorem of Wall [37], any two thickenings of a 1connected, q-dimensional CW -complex in S^{n} are concordant in S^{n} whenever 2q ≤ n − 1.

The functions we are looking for will be constructed by combining the results above with certain homotopical arguments.

It is useful to introduce ”finite” variants of the notion of cone-length. The n-finite, relative cone-length Clf (X, Y)^{n} is defined as in Definition 4 with the additional restriction that each Z_{i} is finite of dimension bounded by n. When the map, f : Y ⟶ X, used to define the relative cone-length, iskconnected define the k-connected, n- finite, relative cone-length, Clf (X, Y)_{k}^{n} by adding the condition that the spaces Z_{i} are (k − 1)-connected. We denote Clf (X)=Clf (X, ∗).

Corollary 3.7 Suppose (W^{(n+1)}, V_{0}, V_{1}) is a triad with W, V_{0}, V_{1} 2- connected and n ≥ 5. There is a smooth triad function on W with only Clf(X, Y)_{2}^{(n−1)/2} reasonable critical points.

Proof. Fix a cone decomposition: Z_{i} ⟶ X_{i} ⟶ X_{i+1} with 0 ≤ i < t =Clf (X, Y)_{2}^{(n−1)/2} where X_{0} = Y, X_{m} = X and Z_{i} are finite, simply connected and of dimension ≤ (n − 1)/2. The dimensionality restrictions imply that we can construct a decomposition of W in triads (W_{i}; T_{i−1}, T_{i}) such that T_{0} = V_{0}, T_{m} = V_{1}, W_{1} ≃ X_{1}, W_{1} ⋃_{T_{1}} W_{2} ⋃ ... ⋃_{T_{i−1}} W_{i} ≃ X_{i}. The theorem provides triad functions g_{i} : W_{i} ⟶ R such that g_{i} has only one critical point which is of A-index the homotopy type of Z_{i−1}. We conclude by pasting the g_{i}’s to get a single triad function on W with the desired properties.

The statement of the corollary shows that the relative cone length and the related invariants are very well suited to our problems. Unfortunately they are quite difficult to compute. In particular, the relative cone-length does not relate well to the relative L.S.-category. On the other hand, in the absolute case, we have a very good relationship between the two notions and this makes possible the following result.

We will say that a manifold M is homotopically of dimension n if it is homotopy equivalent to a n-dimensional, finite CW -complex.

Theorem 3.8 Let M^{2m} be a compact, 2-connected manifold with simply connected boundary. If M is homotopically of dimension m, then there is a smooth function f : M ⟶ R which is constant, maximal and regular on ∂M such that f has only reasonable critical points, P_{i}, with −1 ≤ i <cat(M)+1, f(P_{i}) = i and P_{i} is of index at least i for i ≥ 0, P_{−1} being a minimum.

Proof. Suppose that M ≃ X, where X is a finite CW -complex of dimension m. We first prove a homotopical statement.

Lemma 3.9 Assume X finite and k-connected. There is a cone decomposition Z_{i} ⟶ X_{i} ⟶ X_{i+1}, 0 ≤ i <cat(X)+1, such that dim(Z_{i}) ≤ m − 1, Z_{i} is (k-1)-connected, finite and an i-th order suspension.

In particular cat(X)≤Cl(X)≤Clf(X)_{k}^{m−1} ≤cat(X)+1 .

Proof of the lemma. Let cat(X)= n. By [3] there are cofibration sequences: Z_{i} ⟶ X_{i} ⟶ X_{i+1} with 0 ≤ i < n, X_{0} ≃ ∗, X_{n} ≃ X ∨ ΣT with T ≃ ΩE_{n}ΩX and Z_{i} ≃ (E_{i}ΩX × T)/T. Here E_{i}ΩX ⟶ B_{i}ΩX ⟶ X is the i-th approximation in the Milnor classifying construction [21] applied to ΩX. Recall that E_{i}ΩX ≃ (ΩX)^{∗(i+1)} hence the connectivity of the Z_{i}’s is at least that of X less one. Also note that Z_{i} ≃ Σ^{i}U_{i} where U_{i} ≃ ((ΩX)^{∗(i+1)} ×T)/T. Notice also that ΣT is in fact an n-suspension. Assume now that we have a certain cell decomposition for U_{i} and let U_{i}^{(m−i−1)} be the corresponding skeleton. Clearly, Z_{i}^{(m−1)} = Σ^{i}U_{i}^{(m−i−1)} is an (m − 1)-skeleton of Z_{i}. We construct inductively commutative diagrams:

[figure: Z_{i}^{(m−1)}; X_{i}^{(m)}; X_{i+1}^{(m)}; Z_{i}; X_{i}; X_{i+1}]

with the rows homotopy cofibration sequences and the two right vertical maps each inducing an epimorphism in homology in dimensions less or equal to m and an isomorphism in dimensions less or equal to m − 1. Consider a cell decomposition of Q = ΣT such that Q^{(m)} is an n-th order suspension and the lifting of the map X_{n}^{(m)} ⟶ X ∨ Q to a map X_{n}^{(m)} ⟶ X ∨ Q^{(m)} induces an epimorphism in homology which is an isomorphism in dimensions strictly less than m. This means that there is a wedge of m dimensional spheres, V, such that we have a homotopy cofibration sequence: V ⟶ X_{n}^{(m)} ⟶ X ∨ Q^{(m)}. As X ∨ Q^{(m)} is a retract of X_{n}^{(m)} it follows that we can build the following commutative diagram:

[figure: V; V ∨ Q^{(m)}; Q^{(m)}; V; X_{n}^{(m)}; X ∨ Q^{(m)}; ∗; X; X]

where the vertical and horizontal lines are all homotopy cofibration sequences.

This produces the desired cone decoposition if one notes that V ∨ Q^{(m)} is an n suspension (for cat(X)≤ dim(X) always holds).

For the theorem, we use the lemma to produce a ”good” cone decomposition of M. Next, as in Corollary 3.7, we can construct a triad function f′ with critical points P_{i}, −1 ≤ i < t, t ≤cat(M)+1 and such that all these critical points, are reasonable, f(P_{i}) = i, for i > −1 A_{f}(P_{i}) is an isuspension homotopy equivalent to a complex of dimension at most m − 1 and P_{−1} is a minimum. Assume that A_{f}(P_{i}) = Σ^{i}U_{i}. The dimension of U_{i} is at most m − i − 1. That means that it can be thickened inside a 2(m − i − 1) + 1 dimensional sphere. As in Proposition 3.1 we can construct a function g_{i} : D^{2(m−i)} ⟶ R with a reasonable critical point at the origin and with A_{g}(0) ≃ U_{i}. Consider the function g′_{i} : D^{2m} = D^{2(m−i)} × D^{2i} ⟶ R defined by g′_{i}(x, y) = g_{i}(x)+q(y) where x ∈ D^{2(m−i)}, y ∈ D^{2i} and q : D^{2i} ⟶ R is a nondegenerate quadratic form of index i. It follows from Corollary 2.7 that A_{g′_{i}}(0) = A_{g_{i}}(0) ∗ A_{q}(0) = U_{i} ∗ S^{i−1} = A_{f}(P_{i}). We now apply the second part of Theorem 3.4 to replace the function f′ by one that, up to a change of coordinates, coincides with g′_{i} near each critical point P_{i}.

Remarks. 1. It immediately follows from the theorem that in the homotopy type of any 2-connected, finite CW -complex X there is a compact manifold with boundary M that supports a function with no more than cat(X)+2 critical points. For 2-connected spaces, this is the best possible result. Indeed this follows by considering Corollary 2.5 and by recalling that there are finite, 2-connected spaces of category one (= co-H-spaces) that are of cone-length two.

Of course the theorem is much more precise than that. The constructed function is self-indexed and the dimension of the manifold can be assumed to be at most twice that of X.

2. Notice that if 0 is a critical point of index i for the function f : D^{n} ⟶ R then, by a parametrized Morse lemma [13], f decomposes locally as a sum g(x_{1}.....x_{k}) + q(x_{k+1}.....x_{n}) with q a non-degenerate quadratic form of index i.

4 Final comments

1.Critical points of A-index a wedge of spheres.

We have seen in Proposition 3.5 that under certain circumstances we can ”fuse” critical points. In some cases this is much simpler to do.

Consider the triad (W^{n+1}; V_{0}, V_{1}) such that V_{0} is connected and W^{n+1} is obtained from V_{0} by the disjoint attachment of a finite number of handles (we mean by this that the embeddings of these handles in W may be assumed to have disjoint images). Then there is a smooth left cofibration Z ⟶ V_{0} ⟶ W with Z a homotopy wedge of spheres. Indeed, as V_{0} is connected all we have to do is to connect all the disjoint handles by a tubular neighbourhood of a path in V_{0}.

In particular, if on W there is a Morse triad function with all critical points of the same index and belonging to the same level, then all these critical points can be fused to a single degenerate one. An alternative proof of this can be found in [34]. The same statement remains true if W, V_{0}, V_{1} are simply-connected and the indices of the critical points are ≤ (n − 1)/2.

Of course, the A-index of a degenerate critical point constructed in these cases is a wedge of spheres.

One way to apply these remarks is to notice that:

Lemma 4.1 Assume the context of Theorem 3.8 except that we only require M to be simply connected, then there is a triad function on M with cat(M)+3 critical points.

Proof. The result is trivial for m = 2. As in the proof of the Theorem 3.8 we asssume that M ≃ X with X a finite CW -complex of dimension m ≥ 3. In the cofibration sequences giving a cone-decomposition of X, the finite spaces Z_{i} are simply connected for i ≥ 1, however Z_{0} might not be so (see Lemma 3.9). Let Z′ be its one dimensional skeleton. As it is connected it is homotopy equivalent to a wedge of, say, t circles. By inspecting the diagram of cofibrations:

[figure: Z′; Z′; ∗; Z_{0}; ∗; X_{1}; Z_{0}/Z′; ΣZ′; X_{1}]

we see that if we build inside M a triad M_{0}^{(2m)} ≃ ΣZ′ with ∂M_{0} simply connected, together with a function f_{0} : M_{0} ⟶ R with only two critical points which are both reasonable and which is maximal, constant and regular on the boundary, we can continue by applying Corollary 3.7 to the triad M − M_{0}. For M_{0} we embed in M the triad(S^{2} × D^{m−2})^{#t}. The construction of f_{0} is immediate.

We next see that, if we use rational homotopy equivalence instead of homotopy equivalence, critical points having for A-indices wedges of spheres are enough for obtaining the minimal possible number of reasonable critical points. We refer to [14,35] for the basics of rational homotopy. In particular, a simply connected space is rational when all its homotopy groups are rational vector spaces. There is a rationalization functor from simply connected, finite type CW -complexes to rational spaces. The rationalization of such a CW - complex X is denoted by X_{(0)} and there are maps X ⟶ X_{(0)} inducing an isomorphism in rational homology. Any such map is called a rationalization map. The rationalization of any suspension is homotopy equivalent to the rationalization of a wedge of spheres. The rational homotopy type of a space is the homotopy type of its rationalization. Finally, the rational homotopy type of a space X is completely encoded in the homotopy type of its (rational) algebra of picewise linear forms A_{P L}(X) (this is considered in the category of augmented, differential, graded, rational algebras).

All the definitions of our numerical homotopy invariants make sense in the category of rational spaces.

Lemma 4.2 Assume X is a simply connected, finite CW -complex. There is a manifold M of the rational homotopy type of X that supports a smooth function f : M ⟶ R regular, constant and maximal on the boundary of M and that has no more than Cl(X_{(0)})+1 critical points which are all reasonable of A-indices wedges of spheres.

Proof. The lemma follows from the remarks above and the two statements bellow:

Fix a simply connected, finite type space B of rationalization B_{(0)}. The finite cone-length and the cone-length of B_{(0)} coincide. Indeed, from [4] we know that the cone-length of B_{(0)} equals the homotopy nilpotency of the algebra A_{P L}(B), that is the smallest natural m such that in the homotopy type of A_{P L}(B) there is a differential, graded, commutative, augmented, rational algebra B with (\overline{B})^{m+1} = 0. Suppose now that the dimension of B_{(0)} is n. This implies that there is an ideal J of B which is acyclic, contains all the elements of degree greater or equal than n + 1 and consists only of elements of degree at least n. Let B′ = B/J. Clearly, this new algebra is finite and it is nilpotent of the same order as B. There is a standard way [4] to associate to B′ a cone decomposition of a space of the homotopy type of B_{(0)} such that at each stage the cone is attached over a finite wedge of spheres whose dimension is bounded by one less that of B′ and the length of the decomposition is equal to the order of nilpotence of B′.

Any cell decomposition of B_{(0)}, can be delocalized. In fact, let S_{(0)}^{n} ⟶^{α} A_{(0)} ⟶ C_{(0)} be a rational cell-attachement with n ≥ 2 with A_{(0)} rational, simply connected and suppose that there is a rationalization map l : A ⟶ A_{(0)}. There is α′ ∈ π_{n}(A) such that l(α′) = kα for some integer k. It is easy to see that the cofibre of kα and that of α have the same rational homotopy type.

As the rationalization of a suspension has the homotopy type of a wedge of rational spheres, for a simply connected, finite, rational space, X_{(0)}, we deduce the existence of a CW -complex, Y, rationally equivalent to X_{(0)} and which admits a cone decomposition: W_{i} ⟶ Y_{i} ⟶ Y_{i+1}, 0 ≤ i <Cl(X_{(0)})= m with Y_{0} = ∗, Y = Y_{m} and W_{i} a finite (i − 1)-connected wedge of spheres of dimension strictly smaller than that of X_{(0)}.

Remark. It is interesting to note the existence of a conjecture of Lemaire and Sigrist [19] that claims that, in the rational context, the cone-length and the L.S.-category coincide.

2.Closed manifolds.

Our methods can also be applied to the case of closed maifolds. There is a further type of homotopical considerations that we need for this purpose. Suppose X is a space with the homotopy type of a finite CW -complex. Let X^{(i)} be an i-th dimensional skeleton of X. This is a CW -complex of dimension i such that there is a map X^{(i)} ⟶ X inducing an isomorphism in homology in dimensions smaller or equal to i − 1 and an epimorphism in dimension i. As in lemma 3.9 it is easy to see that among all such X^{(i)}’s, the L.S.-category (and, similarly, the cone-length and the finite cone-length) can vary by at most one. We denote the maximum of these values by cat^{i}(X) (and similarly for the cone-length and the finite cone-length).

Lemma 4.3 Let M be a closed, 2-connected n-manifold. There is a smooth function f : M ⟶ R with only reasonable critical points in number at most 2cat^{[n/2]}(X)+5.

Proof. Consider on M a Morse function g with just one minimum and just one maximum and such that all critical points of index j are of critical value j > 1. Consider first the case when n = 2k. It is possible to write M = M_{1} ⋃_{V} M_{2} with M_{1} and M_{2} n-manifolds with boundary difffeomorphic to V both 2-connected and such that M_{1} is, up to homotopy, a k- skeleton of M and M_{2} a k − 1-skeleton. Indeed, we take V = g^{−1}((2k + 1)/2) and M_{1} the side of V containing the minimum and M_{2} the side containing the maximum.

Now we apply Theorem 3.8 to M_{1} and M_{2} thus obtaining on M_{1} a function with at most cat(M_{1})+2 reasonable critical points and similarly for M_{2}. Notice that cat(M_{2})≤cat(M_{1})+1. The result now follows by pasting one of the functions above to the inverse of the other.

If n = 2k + 1 we write M = W_{1} ⋃ W_{2} ⋃ W_{3} the first union being taken following g^{−1}(k + (1/2)) and the second following g^{−1}(k + (3/2)). We note that W_{1} and W_{2} are both k-skeleta of M. All the critical points of index k + 1 can be pushed to a single degenerate one.

Remark. It is obvious that in many particular cases one can do better than the result above. In fact, the main deficiency of this result comes from the fact that 2cat^{[dim(M)/2]}(X) is not a good approximation for cat(M). Indeed, consider (S^{2} × S^{2} × S^{2}) × S^{7}.

3. Functions with only three reasonable critical points on closed manifolds.

The basic idea that we defend here is that if degenerate, reasonable critical points are more complicated than non-degenerate ones they also contain more global information concerning the ambient manifold. A number of interesting facts on this topic appear in [34].

We start our discussion here with an elementary, but instructive, discussion. Let I = [0, 1]. We consider I oriented in the direction going from 0 to 1. Consider two oriented 2-dimensional disks D_{1} and D_{2}. Embedd in ∂D_{1}, (and respectively, in ∂D_{2}) three disjoint segments I_{1}, I_{2}, I_{3} (and respectively, I′_{1}, I′_{2}, I′_{3} such that the orientation is preserved. Join I_{i} to I_{j} by a Möbius band. That is take I × [0, 1] and paste I × {0} to I_{i} and I × {1} to −I′_{i}. Let W be the resulting 2-manifold with boundary. It has a connected boundary and if we paste a disk following ∂W what we get is a torus. This shows that on the torus there is a function with exactly three critical points. Two of the critical points are simple: a maximum and a minimum. Concerning the third, say P, the same arguments as those used in Lemma 3.3 show that, up to a change of coordinates, in its neighbourhood the function can be taken of the form f = Re(z^{3}). The same type of construction can be done in the case of an orientable surface of genus g. In that case the local form of the function around P can be assumed to be f = Re(z^{2g+1}).

Notice that on an orientable surface we can not construct a function that has the form f = Re(z^{2k}) in the neighbourhood of P. However, this can happen on a nonorientable surface (on RP^{2} there is a Morse function with three critical points). Notice also that, in all cases, the singular hypersurface f = 0 cuts a small sphere around P in an even number of points. These are two facts that we will generalize below.

Remark. It follows from the above that if f : (W^{(n+1)}, V_{0}, V_{1}) ⟶ R is a triad function that has exactly t critical points all of index k which are all on the same critical level, then we can replace f by another triad function g with just one critical point P such that, up to a change of coordinates, g = Re(z^{t+1}) + q where q is nondegenerate of rank n − 1 and index k − 1.

Consider a closed, (n + 1)-dimensional manifold M, n > 4, and a smooth function f : M ⟶ R such that f has exactly three critical points: a maximum, a minimum and a third, P, which is reasonable.

Lemma 4.4 Let T_{ϵ} = f^{−1}(f(P)) ⋂ S(P, ϵ) where S(P, ϵ) is the sphere centered at P and of radius ϵ. If ϵ is as in Lemma 1, then for (n − 1) > i > 0 we have H_{i}(T_{ϵ}; Z) ≃ H_{i+1}(M; Z) ⨁ H_{i+1}(M; Z) and the number of connected components of T_{ϵ} is equal to 2rk(H_{1}(M; Z)) − 1.

Proof. As f has only three critical points we have to analyze the triad (N; V_{0}, V_{1}) with V_{0} and V_{1} homeomorphic to n-dimensional spheres, f : N ⟶ R having as single critical point P. Here N is obtained from M by removing two disks which contain respectively the maximum of f and the minimum of f. Notice that the n-dimensional skeleton of M, X, is obtained from N by attaching a disk along V_{0} (or, equivalently, by removing an (n + 1)-disk from M). The homotopy cofibration sequence: A_{f}(P) ⟶ V_{0} ⟶ N shows that X′ ≃ ΣA_{f}(P). Let Z be the manifold with boundary (constructed in the proof of the Theorem 2.4) which represents A_{f}(P). Let Q be the analogous manifold corresponding to A_{−f}(P). Notice that V_{0} − Int(Z) and V_{1} − Int(Q) are homeomorphic manifolds with boundary. By excision, this shows that for all 0 ≤ i < (n − 1), H_{i}(Z; Z) ≃ H_{i}(Q; Z) . By noting that Z ⋃_{T_{ϵ}} Q ≃ S^{n} and considering the Mayer-Vietoris exact sequence we see that H_{i}(T_{ϵ}; Z) ≃ H_{i}(Z; Z) ⨁ H_{i}(Q; Z) for 0 < i < (n − 1). The second statement is obvious.

Lemma 4.5 Suppose that, with the notations above, there is a system of coordinates around P with respect to which f is even. Then χ(M) is odd.

Assume that M is also simply connected and let X be obtained from M by deleting a small, open, (n+1)-dimensional disk. For any odd prime p the localised space X_{(p)} splits nontrivially as a wedge except, possibly, when χ(M) is equal to 1 or 3.

Proof. First observe that, as f is even around P, we have a free Z/2 action on f^{−1}(−∞, f(P)] ⋂ S(P, ϵ). Indeed f is invariant with respect to the antipodal map on S(P, ϵ). This means that χ(A_{f}(P)) is even. As X ≃ ΣA_{f}(P) the first point follows.

For the second part notice first that Z ≃ f^{−1}(−∞, f(P)] ⋂ S(P, ϵ) is connected. Denote by i : X ⟶ X a map in the homotopy class of the suspension of the antipodal map a restricted to f^{−1}(−∞, f(P)] ⋂ S(P, ϵ). Of course, i^{2} ≃ id. This allows one to split the space X, after localization at an odd prime, by a classical homotopy trick. Consider j = H_{∗}(i) : H_{∗}(X) ⟶ H_{∗}(X). We have j^{2} = id. Hence j can have as eigenvalues only −1 and 1. Let A be the direct limit of X ⟶^{i−id} X.... ⟶^{i−id} X ⟶^{i−id} .. and similarly B the direct limit obtained by using the map i + id instead of i − id. Notice that the obvious map X ⟶ A ⋁ B induces an isomorphism in homology (recall that 2 is inverted). Now, A is trivial if 1 does not appear as an eigenvalue of j other than in dimension 0. As j is the suspension of the antipodal map, a, the same is true for H_{∗}(a). We compute the Lefschetz number of a, λ(a) = −(χ(Z) − 1) + 1. On the other hand we have χ(M) = −(χ(Z) − 1) + 1 + (−1)^{n+1}. Recalling that the antipodal map has no fixed points and that n + 1 is even we obtain λ(a) = 0 and χ(M) = 1. Similarly B is trivial if −1 does not appear as an eigenvalue of j and hence of H_{∗}(a). This time λ(a) = χ(Z). This implies χ(M) = 3.

Remark. There is a Morse function on CP^{2} with exactly three critical points. This shows that the χ(M) = 3 exception can occur. A similar example provides the χ(M) = 1 case.

Two open questions.

1. Is the Conley index of every isolated critical point a suspension ? We believe the answer to be negative.

2. Does every smooth, closed, simply connected 4-manifold support a smooth function with exactly three critical points ?

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