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A Whitney embedding theorem for a class of stratified Frölicher spaces Tore August Kro, tore.a.kro@hiof.no March 7, 2016 Abstract Classically Whitney described stratified sets as subsets of Rn splitting as a disjoint union of submanifolds satisfying conditions A and B. To take a more modern approach to stratified spaces, choose some category of generalized smooth spaces, define a list of allowed types of neighborhoods, and then declare W to be stratified if every point in W has a neighborhood diffeomorphic one of the allowed types. In this paper we employ Frölicher spaces, which is a complete, cocomplete and Cartesian closed category. We allow neighborhoods T = U × CL being the product of an open subset U of some Euclidean vector space and a cone CL. Note that smooth function spaces, for example Frö(W, Rn ), appear automatically as categorical constructions. The aim of this paper is to generalize the Whitney embedding theorem to our class of stratified spaces. We define S-embeddings φ : W → Rn and show that for compact W of dimension m, the set of S-embeddings is open and dense in the mapping space Frö(W, Rn ) when n > 2m. Moreover, the images φ(W ) of S-embeddings are Whitney stratified sets in the classical sense. 1 Introduction The field of differential topology has produced fascinating results about manifolds. Among these are the sphere eversion [Sma58, FM80], smooth structures on the 7-sphere, [Mil56, Wal99] and surveyed in [Lüc02], and the calculation of the mapping class group [MW07, GTMW09]. Stratified spaces generalize the concept of manifolds. Classically, for stratified subsets of manifolds Whitney introduced in [Whi65] his conditions A and B on how tangent planes behave when approaching a lower stratum. Topologically a stratification of X is a cover consisting of pairwise disjoint manifolds of varying dimension. They are usually required to satisfy locally finiteness and the axiom of the frontier, basically describing how the boundary of 1 one stratum is the union of lower dimensional strata. These axioms can be found in [Tho69, Tho71, Mat70]. There are several other axioms for stratified subsets of smooth manifolds, both stronger and weaker than Whitney’s condition B. These regularity conditions are surveyed by Trotman in [Tro07]. There are several classes of topological stratified sets. Siebenmann introduced in [Sie72] locally conelike TOP stratified sets. His definition is analogous to the sharp stratified spaces introduced in this paper, the difference is mainly the category in which the construction takes place. Later Quinn introduced in [Qui88] manifold homotopically stratified sets in order to give a setting for the study of purely topologically stratified phenomena. Hughes and Weinberger survey in [HW01] these topologically stratified sets through applications of intersection homology and surgery theory. In order to do differential topology for stratified spaces one would like to define the notion of a smooth structure on a topologically stratified set and generalize tools such as transversality, Morse theory, the Whitney embedding theorem and many other results. Moreover, there should be new results, such as Thom’s first and second isotopy lemmas [Mat70]. Defining the smooth structure on a stratified space has its difficulties. Mather introduce abstract pre-stratified sets by giving data on how tubular neighborhoods of nearby strata relates. This amount of data leads to a technically demanding theory, and function spaces of smooth maps are not immediately given. Later Mathias Kreck [Kre10] introduced stratifolds by using Sikorski’s differential spaces [Sik72] to keep track of the smooth structure. Unfortunately, for any stratifold S, which is not a manifold, there exists no smooth injective map S ,→ Rn for any finite n. In particular, for the half open interval as a stratifold, any smooth map [0, 1) → R is, by definition, locally constant near 0. Meanwhile, another category of generalized smooth spaces was developed. Boman shows in [Bom67] that a function f : Rn → R is smooth if and only if the composition f ◦ c is smooth for all smooth curves c : R → Rn . Frölicher uses this property to define a category of smooth spaces [Frö82]. This category is now known as Frölicher spaces. A result due to Lawvere, Schanuel and Zame [LSZ81] implies that Frölicher spaces is Cartesian closed. Already Frölicher notes that his category includes objects with singularities, but together with Kriegl and Michor the theory is developed in the direction of infinite dimensional spaces [KM97, FK88]. In this direction Cap [Cap93] developes in his thesis a K-theory for convenient algebras. In [Che00] Cherenack begins to consider the finite dimensional aspects of Frölicher spaces. Cherenack asks in [BC05] if the Frölicher structure on the cone S 1 × [0, ∞) /S 1 × 0, constructed as a pushout, is equal to the subspace structure of a cone embedded in R3 . Another interesting question is the topology on the Frölicher space of maps with compact support, see [Che01]. Dugmore and Ntumba [DN07] does homotopy theory in Frölicher spaces by making use of flattened and weakly flattended unit intervals. While 2 Batubenge and Tshilombo [BT09, BT14] study topologies on final and initial objects, products and coproducts. A nice collection of finite dimensional examples appear in [HKM14]. The development of Frölicher spaces has recently been surveyed by Batubenge [Bat15]. Moreover a comparative study by Stacey [Sta11] relates Frölicher spaces to other alternative categories of generalized smooth spaces. In this paper we define our class of stratified spaces by specifying what types of neighborhoods that are allowed. A sharp stratified space W is basically a Frölicher space where each point has a neighborhood T , called a tubular domain. These tubular domains are products T = U × CL, where CL is a cone and U an open subset of some Euclidean space. The main result is a Whitney embedding theorem for compact sharp stratified spaces. It is not hard to generalize this local approach and define larger classes of stratified spaces. Prominent examples of types of neighborhoods that could be included are orbit spaces V /G, where G is a compact Lie group acting smoothly on a finite dimensional vector space V . However, the aim in this paper is not to define the most general notion of a smooth stratified space, but to prove the Whitney embedding theorem. This paper is organized as follows: In section 2 we briefly review the definition of a Frölicher space. We construct sharp stratified spaces in section 3. Our main technical tool is Hadamards lemma 3.5. Moreover, we prove that sharp stratified spaces are topologically stratified. In section 4 we discuss the topology of the mapping spaces. Noting that they are Fréchet spaces, we get an explicit description of the open sets. In section 5 we introduce coordinate systems, the scaled Jacobi matrix, and prove a scaled mean value theorem. In section 6 we discuss the classical Whitney conditions and show that the hold for reasonable images of sharp stratified spaces into Euclidean space. In section 7 we show that the set of S-immersions is open in the mapping space. Furthermore, we show in section 8 that the set of S-embeddings also is open. There are plentiful of S-embeddings. In section 9 we show that the set of S-embeddings is dense if the dimension of the codomain is sufficiently large. Our main result is: Theorem 1.1 Given a compact sharp stratified set W of dimension m. If n > 2m, then there is an open and dense set of mappings φ : W → Rn in the function space Frö(W, Rn ) such that φ is injective and the image φ(W ) is a Whitney stratified set. 2 About Frölicher Spaces This section is a concise summary of Frölicher spaces. We recall the definition of the category Frö, and review how it is a complete, cocomplete and Cartesian closed category. Boman’s theorem shows that smooth manifolds 3 are contained in Frö as a full subcategory. Furthermore, we discuss topologies on a Frölicher space. The half open interval is of particular interest as it is used in the construction of tubular domains. Definition 2.1 A Frölicher space X is a triple (X, CX , FX ) where X is a set, CX is a subset of Map(R, X) and FX is a subset of Map(X, R) such that c : R → X lies in CX if and only if f c ∈ C ∞ (R, R) for all f ∈ FX , and f : X → R lies in FX if and only if f c ∈ C ∞ (R, R) for all c ∈ CX . The pair (CX , FX ) is the smooth structure on X. We call CX and FX the smooth curves and smooth functions respectively. A map φ : X → Y between the underlying sets of two Frölicher spaces is called smooth if for each f ∈ FY , the pullback f φ lies in FX . Let Frö denote the category of Frölicher spaces and smooth maps. Clearly the composition of two smooth maps is again a smooth map. Moreover, (1) saying that φ : X → Y is smooth is equivalent to (2) saying that for all c ∈ CX we have φc ∈ CY , and (3) saying that for all c ∈ CX and f ∈ FY the composition f φc lies in C ∞ (R, R). The real numbers R form a Frölicher space by declaring CR = FR = ∞ C (R, R). The n-dimensional Euclidean space Rn is a Frölicher space where the smooth curves are C ∞ (R, Rn ) and the smooth functions are C ∞ (Rn , R). Boman’s theorem [Bom67] verifies the hard part of Definition 2.1 in this case, namely that f : Rn → R is smooth if the composed function f c lies in C ∞ (R, R) for every smooth curve c in Rn . The category Frö is both complete and cocomplete, see [FK88, Corollary 1.1.5]. A smooth structure (CX , FX ) on X is finer than another smooth 0 , F 0 ) on the same set if C ⊆ C 0 (or equivalently F 0 ⊆ F ). structure (CX X X X X X Given any set C0 ⊆ X R there is a finest smooth structure on X such that C0 ⊆ CX ; we call it the smooth structure generated by C0 and we have FX = {f : X → R | f c ∈ C ∞ (R, R) for all c ∈ C0 }. Similarly, given any set F0 ⊆ RX there is a coarsest smooth structure on X generated by F0 , i.e. F0 ⊆ FX . Consequently, the forgetful functor U : Frö → Set has both a left and a right adjoint. It follows that U preserves both limits and colimits, and the underlying set of a (co)limit is the (co)limit of the underlying sets. To prove that Frö is Cartesian closed one needs to choose a smooth structure on C ∞ (R, R) satisfying certain conditions, see [Frö82] and [FK88, Theorem 1.1.7]. The condition is verified in [FK88, Theorem 1.4.3]. The curves c in a smooth structure on C ∞ (R, R) can be defined precisely as the maps 4 c : R → C ∞ (R, R) such that the adjoint c̃ : R × R → R is smooth. Equivalently, c is smooth if and only if it is a smooth curve c : R → C ∞ (R, R) when C ∞ (R, R) is considered as the Fréchet space with the topology of uniform convergence of each derivative on compact sets, see [KM97, Theorem 3.2]. More generally, the smooth structure on the internal hom Frö(X, Y ) is defined to have smooth curves given by CFrö(X,Y ) = {c : R → Frö(X, Y ) | the adjoint c̃ : R × X → Y is smooth}. As a direct consequence of being Cartesian closed, it follows that taking cross product with a fixed Frölicher space commutes with taking colimits. In general there may be several interesting topologies on a Frölicher space X. As noted by Cap in his thesis [Cap93, Paragraph 1.9], there are two extremes. We always want R to have the standard topology. Using Stacey’s names for these topologies, see [Sta13, Definition 2.1], we define the functional topology on X as the coarsest topology such that all smooth functions f : X → R are continuous. The curved topology on X is the finest topology such that all smooth curves c : R → X are continuous. The curved topology is called the c∞ -topology in [KM97, Definition 2.12]. For some X the two extreme topologies are the same, and in that case X is usually called balanced following Cap’s terminology, or smoothly regular by Stacey. All manifolds are balanced. When X is a function space, there are in general several different topologies, see [KM97, Example 4.8]. A Frölicher space X is called Hausdorff if the functional topology, hence also the curved topology, on X is Hausdorff, i.e. if for any pair of distinct points p1 , p2 in X there is a smooth function f ∈ FX with f (p1 ) 6= f (p2 ). Similarly, we define a Frölicher space to be locally compact, compact, paracompact, or to have a countable basis for its topology, etc. if this holds for its functional topology. Proposition 2.2 A paracompact Frölicher space admits smooth partitions of unity subordinate to any open cover. Proof: Let {Uα } be an open cover of a paracompact Frölicher space X. By definition of the functional topology on X we can cover X with smooth nonnegative functions fβ such that the support of each fβ is contained in some Uα . Pass to a locally finite refinement of the family of functions {fβ }. For each Uα construct a new function by taking the sum of all fβ supported in Uα . Normalization gives a smooth partition of unity subordinate to {Uα }. On the half open interval [0, ∞) we take the Frölicher subspace structure corresponding to the inclusion [0, ∞) ⊆ R, i.e. C[0,∞) are all curves c : R → [0, ∞) such that the composition ⊆ c R→ − [0, ∞) − →R 5 is smooth. Bentley and Cherenack [BC05, Proposition 3.1] characterize the smooth functions F[0,∞) as functions f : [0, ∞) → R which are smooth at all interior points of [0, ∞) and right smooth at 0. This follows from the more general fact that Frölicher smooth functions on a codimension 0 smooth submanifold M with boundary of a manifold E extends to a smooth function on an open neighborhood of M in E, see [KM97, Theorem 24.8]. On the other hand, we can form the quotient R/± in the category of Frölicher spaces by identifying t ∈ R with −t ∈ R. Bentley and Cherenack prove that the map η : R/± → [0, ∞) given by η(x) = x2 is a diffeomorphism of Frölicher spaces. There are several other smooth structures on the half open interval. Dugmore and Ntumba [DN07] define, for the purpose of doing homotopy theory, a flattened and a weakly flattened unit interval. However, we will not use these structures in this paper. 3 Construction Being a stratified space should be a local property. One approach to their construction is thus given by specifying a collection of allowed neighborhood types, and defining a stratified space as a Frölicher space locally diffeomorphic to one of these neighborhoods. Actually this construction can also proceed in several steps, starting with the collection of all finitely dimensional Euclidean spaces. The sharp stratified spaces will be constructed below by inductively considering certain pushouts over the half open interval as allowed neighborhoods. Certainly this could be generalized to other types of stratified neighborhoods, but the aim of this article is not to give the most general construction of stratified space imaginable, but rather to reproduce Whitneys embedding theorem. We will call our allowed neighborhoods tubular domains. They are defined recursively, but for the initial step the space denoted by L below is just a compact manifold. Definition 3.1 A tubular domain T of height h is the pushout in Frölicher spaces given in the diagram U × {0} × L / U × [0, ∞) × L / T, U where U is an open connected subset of some Euclidean space Rm and L is a height h − 1 sharp stratified space. We call U the local stratum, L the 6 link and BT = U × [0, ∞) × L the blowup. There are natural smooth maps π : T → U and ρ : T → [0, ∞), and we call them the tubular projection and the tubular function respectively. We use the adjectives latitudinal, radial and longitudinal when referring to the directions of U , [0, ∞) and L respectively. Whenever 0 ∈ U we define the base point in T as the image of 0 under the inclusion U → T at ρ = 0. Having specified the allowed neighborhoods, we define sharp stratified spaces of height h as any Frölicher space that locally is diffeomorphic to a tubular domain of height at most h. In addition we demand some topological standard requirements. Definition 3.2 A height h sharp stratified space W is a Frölicher space such that every point p in W has an open neighborhood V diffeomorphic to a tubular domain T of height less than or equal to h, some point p in W has an open neighborhood V diffeomorphic to a tubular domain T of height exactly h, and W is Hausdorff, paracompact, and has a countable basis for its topology. Here are a few examples of sharp stratified spaces: Example 3.3 i) Let M be any compact manifold. The cone of M is a sharp stratified space. ii) Let L be a discrete set with three points, {1, 2, 3}. Let R be the local stratum of a tubular domain T . Take any permutation σ of L. Define W as the quotient space of T where (x, ρ, i) is identified with (x + 1, ρ, σ(i)). iii) The figure eight is a sharp stratified space where the midpoint has a neighborhood diffeomorphic to the cone over four points. iv) The Whitney umbrella can be constructed as a sharp stratified space by taking the cone with link L the union of a figure eight and a disjoint point. Proposition 3.4 Sharp stratified spaces are balanced. Proof: We must show that the functional and curved topologies are identical. By induction assume that all links L are balanced. It follows from [Cap93, Proposition 1.16] that any blowup BT = U × [0, ∞) × L also 7 is balanced. Let V be a curved open set in W , and let x0 be a point in V . It is enough to show that V is a functional neighborhood of x0 . Choose a tubular domain T based at x0 . Let q be the composition BT → T ⊆ W . Then q −1 (V ) is curved open, and by the induction hypothesis also functional open. For u ∈ U , ρ ∈ [0, ∞) and z ∈ L define a smooth map g : BT → R by g(u, ρ, z) = |u|2 + ρ. Choose > 0 such that K = g −1 ([0, ]) becomes a compact subset of BT . Since K r q −1 (V ) is compact in the functional topology the function g has a minimum value, say δ, on this set. Since 0 × 0 × L = g −1 (0) ⊆ q −1 (V ) we have δ > 0. Also δ ≤ . Consequently, g −1 (−δ, δ) is contained in q −1 (V ). Observe that g descends to a well-defined function f : T → R. Now x0 ∈ f −1 (−δ, δ) ⊆ V ∩ T , and this shows that V is a functional neighborhood of x0 in W . Since tubular domains are defined by pushout it is easy to study the smooth functions of T . The main tool will be this version of Hadamard’s lemma. Lemma 3.5 (Hadamard) Given a smooth function f : T → R there are unique functions f0 : U → R and fˆ1 : BT = U × [0, ∞) × L → R such that f = f0 π + ρfˆ1 . Moreover, f0 and fˆ1 depend smoothly on f . By abuse of notation we will often omit π from Hadamards lemma and write f0 instead of f0 π. With this simplification f = f0 + ρfˆ1 . Proof: Let f0 be the composition of the canonical inclusion U → T with f . Observe that g = f − f0 π is a smooth function on T that vanish on the local stratum U . Lift g to a smooth function ĝ defined on the blowup BT . Since g(u, 0, z) = 0 for all z ∈ L the fundamental theorem of calculus gives Z ĝ(u, ρ, z) = 0 Let fˆ1 (u, ρ, z) be on f . R1 1 d ĝ(u, tρ, z)dt = ρ dt ∂ĝ 0 ∂ρ (u, tρ, z)dt. Z 0 1 ∂ĝ (u, tρ, z)dt. ∂ρ By construction f0 and fˆ1 depend smoothly For manifolds there are several ways of defining the tangent space. Alternatives include using curves, derivations or local one-parameter families of diffeomorphisms. These definitions generalize to our setting, but no longer do they produce equivalent notions of tangent space. In order to tell which stratum a point belongs to we will elaborate the curve approach to tangent spaces. They are called the kinematic tangent spaces in [Sta13, Section 4]. 8 Definition 3.6 The curve space Curvex X of a Frölicher space X at the point x is the equivalence classes of smooth curves c : X → R with c(0) = x under the relation that c1 ∼ c2 if (f c1 )0 (0) = (f c2 )0 (0) for all smooth functions f . Proposition 3.7 Let W be a sharp stratified space. For all x ∈ W there exists an integer m such that the curve space Curvex W is homeomorphic to Rm . Moreover, such x lies in the local stratum of a tubular domain with dim U = m. Proof: Let us first prove that any point x ∈ W lies in the local stratum of some T . By induction on height we assume this to hold for any sharp stratified space L of lower height. Any point x ∈ W lies in some tubular domain T . If x ∈ U we are done. Otherwise we may write x = (u, ρ, z) where ρ > 0 and z ∈ L. By the induction hypothesis there is a tubular domain T1 in L such that z lies in the local stratum U1 . Then T contains a tubular domain U × (0, ∞) × T1 where x lies in the new local stratum U × (0, ∞) × U1 . Now assume that x lies in the local stratum U of some tubular domain T . We will identify the curve space Curvex W with the tangent space of U at x. We compare a general curve c to its projection into the local stratum πc. Write f = f0 + ρfˆ1 by Hadamard. We have f0 c = f πc. Let C be a compact neighborhood of x in U and let δ > 0. Since L is compact there is an upper bound K on fˆ1 restricted to C × [0, δ] × L. For sufficiently small t the curve c(t) lies under C × [0, δ] × L and it follows that |f c(t) − f πc(t)| ≤ K · ρc(t). Therefore f c(t) − f πc(t) ≤ lim K· ρc(t) = K·(ρc)0 (0). 0 ≤ |(f c) (0)−(f πc) (0)| = lim t→0 t→0 t t 0 0 Since ρ ≥ 0 we have a minimum for ρc(t) at t = 0. Hence (ρc)0 (0) = 0 and consequently (f c)0 (0) = (f πc)0 (0) for all smooth f . It follows that c and πc are equivalent in the curve space. Thus Curvex W ∼ = Tx U . There are a few axioms related to stratifications of topological spaces. See [Mat70]. A stratification of X is a cover by pairwise disjoint smooth manifolds {Si }. We say that the stratification is locally finite if each point of X has a neighborhood that meets at most finitely many strata. The axiom of the frontier is satisfied if the topological boundary of a stratum Si in X is the union of other strata. Let us use the dimension of curve spaces to define strata. Let dim : W → N be the discrete function sending x to dim Curvex W . 9 Definition 3.8 A stratum S of dimension m in a sharp stratified space W is a connected component of the inverse image dim−1 (m). The dimension of W is defined as the maximal m such that W has a non-empty stratum of dimension m. Proposition 3.9 The strata {S} of W is a stratification. It is locally finite and satisfies the axiom of the frontier. Proof: There is a dimension at each point, so the stratification covers W . Each stratum S is a manifold since given x ∈ S we can choose a tubular domain T , where x lies in the local stratum, and then U ⊆ S is a locally Euclidean neighborhood of x. By induction on height we assume that the stratification of sharp stratified spaces of height lower than W is locally finite. To test locally finiteness for W we will argue that any T meets finitely many strata. The strata of T are U together with U × (0, ∞) × Ŝ for each stratum Ŝ of the link L. But L is compact, so the induction hypothesis implies that L has only finitely many strata. Thus T meets only finitely many strata of W . Let S be a fixed stratum and consider another stratum S 0 that meets the boundary of S. For a contradiction assume that S 0 also contains points outside the boundary of S. Since S 0 is connected there is some tubular domain T such that U ⊆ S 0 and U contains points both inside ∂S and outside ∂S. But this is impossible since the intersection T ∩S has the form U ×(0, ∞)×Y where Y is a union of strata in the link L. Consequently S 0 is completely contained in the boundary of S. This proves the axiom of the frontier. 4 The topology of mapping spaces In this section we will explicitly describe the curved topology of Frö(W, Rn ). We will use a theorem by Frölicher and Kriegl [FK88, Theorem 6.1.4], also [KM97, Theorem 4.11]. It says that given a Fréchet space structure such that the Fréchet smooth curves are the Frölicher smooth curves, then the Fréchet topology equals the c∞ -topology, i.e. the curved topology. Recall that a Fréchet space E is a vector space with a countable family of seminorms, k−kn , such that E is Hausdorff and complete, see [Ham82]. If U ⊆ Rk is open, then the space of smooth functions C ∞ (U, Rn ) is a Fréchet space with seminorms kφkα,K = sup k∂ α φ(x)k, x∈K where K runs through a countable family of compact subsets covering U , and ∂ α denotes the higher order derivative corresponding to the multi-index 10 α. By Cartesian closedness, see [KM97, Theorem 3.12], a curve c : R → C ∞ (U, Rn ) is smooth as a map into a Fréchet space if and only if the adjoint c̃ : R × U → Rn is a smooth map. Consequently, the Fréchet space structure on C ∞ (U, Rn ) agree with the Frölicher space structure. Let us inductively define seminorms on Frö(W, Rn ). By induction on height of W , we assume that there is a Fréchet space structure on Frö(W 0 , Rn ) for all W 0 of height less than W . Definition 4.1 Let T ⊆ W be a tubular domain. Let k − ki be a seminorm on C ∞ (U, Rn ) and let k−kj be a seminorm on Frö(BT , Rn ). Then we define n seminorms k−kπT,i and k−kB T,j on Frö(W, R ) by using the Hadamard lemma n mapping φ ∈ Frö(W, R ) by the composition Frö(W, Rn ) → Frö(T, Rn ) ∼ = C ∞ (U, Rn ) × Frö(BT , Rn ) to a pair (φ0 , φ̂1 ). With this notation we define kφkπT,i = kφ0 ki and kφkB T,j = kφ̂1 kj . The key ingredient in establishing a Fréchet structure on Frö(W, Rn ) is comparing structures on two intersecting tubular domains. To do this we need a technical result: Proposition 4.2 Let T and T 0 be tubular domains in W both based at x0 . There exists a compact neighborhood C of 0 in U , δ > 0 and a smooth map η : C × [0, δ] × L → U 0 × [0, ∞) × L0 commuting with projection into W . Moreover, the ratio between the tubular functions ρr is a strictly positive smooth map on C × [0, δ] × L. Proof: We choose the compact neighborhood C and δ > 0 such that the image of C × [0, δ] × L in W lies inside T 0 . Away from the local strata the identity of W defines η smoothly from C × (0, δ] × L to U 0 × (0, ∞) × L0 . In order to extend η to a smooth map defined on C × [0, δ] × L we have to show that the three projections ηlat : C × [0, δ] × L → U 0 ηrad : C × [0, δ] × L → [0, ∞) ηlong : C × [0, δ] × L → L0 exist and are smooth. The first two projections, ηlat and ηrad exist and are smooth because they are restrictions of the tubular projection π 0 : T 0 → U 0 and the tubular function r : T 0 → [0, ∞) respectively. It remains to show that ηlong extends smoothly. Since the tubular function r : T 0 → [0, ∞) is smooth, we may apply Hadamard’s lemma in T and write r = ρ · r̂, where r̂ is a smooth function 11 defined on C × [0, δ] × L. We claim that r̂ > 0 for all points in its domain. If not, there is a u0 ∈ U and z0 ∈ L such that limn→∞ r̂(u0 , n1 , z0 ) = 0. Since u0 also lies in the local stratum U 0 , we may use Hadamard’s lemma in T 0 near u0 to write ρ = rρ̂, where ρ̂ is smooth on a neighborhood of {u0 } × 0 × L0 in U 0 × [0, ∞) × L0 . Back in C × (0, δ] × L we get the identity ρ = rρ̂ = ρr̂ρ̂ at least for points (u, t, z), with t > 0, sufficiently near (u0 , 0, z0 ). Dividing both sides by ρ we get r̂(u, t, z) · ρ̂(η(u, t, z)) = 1 The link L0 is compact, thus the sequence ηlong (u0 , n1 , z0 ) has an accumulation point z 0 in L0 . Since ρ̂(u0 , 0, z 0 ) is finite, it is an impossibility that limn→∞ r̂(u0 , n1 , z0 ) = 0. This proves the claim that r̂ > 0 everywhere. A priori ηlong is at least defined on C × (0, δ] × L. Let f 0 : L0 → R be a smooth function. We will now show that the composition f 0 ηlong extends to a smooth function on C × [0, δ] × L0 . This implies that ηlong extends smoothly to C × [0, δ] × L0 and will thus complete the proof. Clearly the assignment (u, r, z 0 ) 7→ r · f 0 (z 0 ) defines a smooth function on T 0 . Applying Hadamard’s lemma in T yields r · f 0 = ρ · fˆ where fˆ is a smooth function defined on C × [0, δ] × L. Solving for f 0 ηlong and using r = ρr̂, we get fˆ f 0 ηlong = . r̂ fˆ Moreover, is smooth on C × [0, δ] × L, since fˆ and r̂ are smooth and r̂ > 0. r̂ Thus f 0 ηlong extends uniquely to a smooth function on C × [0, δ] × L, and we are done. For Siebenmann’s locally conelike TOP stratified sets there are examples derived from Milnor’s counterexamples to the Hauptvermutung [Mil61] that show that the link is not unique up to homeomorphism. See the remark in [Sie72] after his Definition 1.2. In the category of Frölicher spaces we may use the technical result above to show uniqueness of links: Corollary 4.3 The link at a point x0 ∈ W is unique up to diffeomorphism. Proof: Given tubular domains T and T 0 based at x0 we get smooth maps η : C × [0, δ] × L → U 0 × [0, ∞) × L0 0 0 0 0 η : C × [0, δ ] × L → U × [0, ∞) × L 12 and above the identity of W . It follows that η 0 is the inverse of η away from the local stratum, but then η −1 = η 0 also in the limit ρ → 0. Restricting over the basepoint x0 gives a diffeomorphism L ∼ = L0 . To construct the Fréchet structure we proceed as follows: Given W , let T be the category of tubular domains, where the local stratum U , link L and map BT → T is part of the structure. The morphisms in T are inclusions T 0 ⊆ T as subsets of W . By induction on height we have for each T a Fréchet space structure on Frö(T, Rn ) ∼ = C ∞ (U, Rn ) × Frö(BT , Rn ). We propose to construct the Fréchet structure on Frö(W, Rn ) as the limit of Frö(T, Rn ) over the category T . Clearly the limit is complete when each Frö(T, Rn ) is complete. The following lemmas check that this construction is well-defined: Lemma 4.4 An inclusion T ⊆ T 0 induce a continuous linear map of Fréchet spaces Frö(T 0 , Rn ) → Frö(T, Rn ). Proof: We can check continuity locally in a neighborhood of some x0 ∈ T . Without loss of generality we may assume that x0 lies in the local strata of both T and T 0 . Thus U maps into U 0 by a local diffeomorphism h : U → U 0 . Using the Hadamard lemma in T 0 and T we get a map C ∞ (U 0 , Rn ) × Frö(BT 0 , Rn ) → C ∞ (U, Rn ) × Frö(BT , Rn ) by sending (φ00 , φ̂01 ) to (φ0 , φ̂1 ), such that the equation φ0 + ρφ̂1 = φ00 + rφ̂01 holds in T . Observe that the map splits as a product. We have φ0 = h∗ (φ00 ) where h∗ : C ∞ (U 0 , Rn ) → C ∞ (U, Rn ) is continuous by the chain rule. To check continuity of φ̂01 7→ φ̂1 we apply proposition 4.2 to get a smooth map η : C × [0, δ] × L → U 0 × [0, ∞) × L0 covering the inclusion T ⊆ T 0 in a neighborhood of x0 . Then φ̂1 is the image of φ̂01 under the composition of three continuous maps: The multiplication by the tubular function r of T 0 is a map r· Frö(BT 0 , Rn ) − → Frör=0 (BT 0 , Rn ), where the target consists of smooth functions vanishing on the local stratum U 0 . By induction on height of W we already know that the map η∗ Frör=0 (U 0 × [0, ∞) × L0 , Rn ) −→ Fröρ=0 (C × [0, δ] × L, Rn ) induced by η is smooth, hence continuous. The last map is the inverse of multiplication by r, and it is smooth, hence continuous, by Hadamard’s lemma: Hadamard Fröρ=0 (C × [0, δ] × L, Rn ) −−−−−−→ Frö(C × [0, δ] × L, Rn ) 13 This proves the lemma. Lemma 4.5 The limit of Frö(T, Rn ) is metrizable, i.e. there is a countable family of seminorms defining the topology. Proof: Consider the functor Frö(−, Rn ) as a sheaf of locally convex complete vector spaces on W . Since W has a countable covering by tubular domains Q {T i }, we get that Frö(W, Rn ) is a closed subspace of the countable product i Frö(Ti , Rn ). Consequently Frö(W, Rn ) is metrizable. Proposition 4.6 There is a countable family of the seminorms on Frö(W, Rn ) of the type described definition 4.1. Thus we have a Fréchet space. Moreover, the smooth curves c : R → Frö(W, Rn ) defined by the Fréchet structure are precisely the Frölicher smooth curves. Consequently, the curved topology on Frö(W, R) equals the Fréchet topology. Proof: The only part remaining of the proof is to compare the Fréchet and Frölicher structures. Let c : R → Frö(W, Rn ) be Fréchet smooth. Consider a tubular domain T and decompose c = c0 + ρĉ1 . By Cartesian closedness for the local stratum U , we get that c0 is Fréchet smooth if and only if the adjoint c˜0 : R × U → Rn is smooth, i.e. c0 is Frölicher smooth. By induction we also have that ĉ1 is Fréchet smooth if and only if it is Frölicher smooth. Running through all T in a covering of W , we get that that Fréchet and Frölicher smoothness coincide. The last statement follows from Frölicher and Kriegl’s theorem, see [FK88, Theorem 6.1.4] or [KM97, Theorem 4.11]. Observe that the seminorms now give an explicit description of neighborhoods in the curved topology on Frö(W, Rn ). 5 Coordinate systems In order to study embeddings of sharp stratified spaces it is crucial to understand the smooth structure in a neighborhood of each stratum. Basically we want to use calculus, so we need coordinate systems. Features in the link shrink when approaching the local stratum, so we will rescale accordingly. We will conclude this section with a scaled mean value theorem. Definition 5.1 A coordinate system u of depth k on W consists of tubular domains (T0 , . . . , Tk ) such that T0 ⊆ W and Tj is a tubular domain in the 14 link Lj−1 for all 1 < j < k. Each local stratum Uj is a subset of some Euclidean space and has coordinates which we will denoted by uj = (uj1 , . . . , ujm ). The blowup of the coordinate system is the Frölicher space Bu = U0 × [0, ∞) × · · · × Uk × [0, ∞) × Lk . There is a natural map Bu → W . In the latitudinal direction we have the lower strata U0 × [0, ∞) × · · · × Uk . The k-th tubular function ρk represents the radial direction of u. The k-th link Lk lies in the longitudinal direction. We say that u is based if 0 ∈ Uj for all j and there is a choice of z0 ∈ Lk . In this case the coordinates of the basepoint are (0, . . . , 0, z0 ). A core of u is for each j a compact neighborhood Cj of 0 in Uj and a δj > 0. Thus we get a compact rectangular subset of coordinates of the form C0 × [0, δ0 ] × · · · × Ck × [0, δk ] × Lk . The image of a core in W is called a patch and will be denoted by P . Let us write ΠU as an abbreviation for the lower strata U0 × [0, ∞) × · · · × Uk . Leaving out the points where some ρj = 0 we write Π̊U for U0 × (0, ∞) × · · · × Uk . Similarly, given a core, we use the notation ΠC for C0 × [0, δ0 ] × · · · × Ck . If we need to emphasize the depth we write Πk U or Πk C. On a coordinate system u of depth k we may apply Hadamard’s lemma repeatedly to any smooth function f . First step gives f = f0 + ρ0 fˆ1 where f0 : U0 → R. Applying the lemma to fˆ1 we get fˆ1 = f1 + ρ1 fˆ2 where f1 : U0 × [0, ∞) × U1 → R is smooth. Inductively we get f = f0 + ρ0 f1 + ρ0 ρ1 f2 + . . . + ρ0 · · · ρk−1 fk + ρ0 · · · ρk fˆk+1 , where fj are ordinary smooth functions of several variables with domain U0 × [0, ∞) × · · · × Uj = Πj U and fˆk+1 is smooth on Bu . Definition 5.2 The Hadamard expansion of f in a coordinate system u of depth k is Hk f = f0 + ρ0 f1 + ρ0 ρ1 f2 + . . . + ρ0 · · · ρk−1 fk . We call ρ0 · · · ρk fˆk+1 the remainder. ∂f ∂uji ∂f and ∂ρ . j ˆ Differentiating a Hadamard expansion, say f = f0 + ρ0 f1 + ρ0 ρ1 f2 , with respect to a coordinate of depth > 0, say u11 , we get For all coordinates, uji and ρj , there are partial derivatives ∂f ∂f1 ∂ fˆ2 = ρ0 + ρ0 ρ1 . ∂u11 ∂u11 ∂u11 15 ∂f Observe that ρ0 can be factorized on the left hand size. This means that ∂u 11 vanish on the stratum where ρ0 = 0. To get information from this partial ∂f derivative at the stratum ρ0 = 0, we will rescale and consider ρ10 ∂u , and 11 similar expressions, instead. Therefore we introduce the scaled gradient and scaled Jacobi matrix. Definition 5.3 Let f be a smooth function of W . With respect to a depth k coordinate system u on W we define the scaled gradient vector to be the vector Sku f with entries 1 ∂f ρ0 · · · ρj−1 ∂uji ∂f 1 ρ0 · · · ρj−1 ∂ρj with j ≤ k and with j < k. Similarly, for smooth maps φ : W → Rn we have the scaled Jacobi matrices Sku φ. Proposition 5.4 The scaled Jacobi matrix Sku is a continuous map defined on Frö(W, Rn ) × Bu . Proof: Consider the depth j − 1 Hadamard expansion with remainder f = f0 + ρ0 f1 + . . . + ρ0 · · · ρj−2 fj−1 + ρ0 · · · ρj−1 fˆj . Only the remainder is non-constant with respect to uji and ρj . Differentiating we get ∂ fˆj ∂f = ρ0 · · · ρj−1 , ∂uji ∂uji ∂ fˆj ∂f = ρ0 · · · ρj−1 . ∂ρj ∂ρj ∂ fˆ and ∂ fˆ Since ∂ujij and ∂ρjj are smooth on Bu and depend smoothly on f in Frö(W, Rn ) the result follows. To match the scaled gradient vector, we will also rescale direction vectors. To do this we need a mean for products of ρ’s. Definition 5.5 Let ρ0 , . . . , ρk and r0 , . . . , rk be non-negative numbers. A mixed product has the form c0 · · · ck where ci is either ρi or ri . The weighted mean of mixed products pk is defined as the integral Z 1 pk = (tρ0 + (1 − t)r0 ) · · · (tρk + (1 − t)rk ) dt. 0 16 Calculations of the first few weighted products are p0 = 12 ρ0 + 12 r0 p1 = 31 ρ0 ρ1 + 16 ρ0 r1 + 16 r0 ρ1 + 13 r0 r1 p2 = 41 ρ0 ρ1 ρ2 + 1 1 1 12 ρ0 ρ1 r2 + 12 ρ0 r1 ρ2 + 12 r0 ρ1 ρ2 1 1 1 + 12 ρ0 r1 r2 + 12 r0 ρ1 r2 + 12 r0 r1 ρ2 + 41 r0 r1 r2 . Now we use these weighted means to rescale the direction vector. Definition 5.6 Let x and y be points in the lower strata ΠU of a depth k coordinate system u. We write x = (v0 , r0 , v1 , . . . , rk−1 , vk ) and y = (u0 , ρ0 , u1 , . . . , ρk−1 , uk ). Define the scaled direction vector Λk (x, y) to be the vector Λk (x, y) = (w0 , λ0 , w1 , . . . , λk−1 , wk ) where wj = pj−1 (uj − vj ) and λj = pj−1 (ρj − rj ) and the scaling factors pj are the weighted mean of mixed products of ρ0 , . . . , ρj and r0 , . . . , rj . Theorem 5.7 (A scaled mean value theorem) Let f : W → R be a smooth function and let u be a depth k coordinate system. Precomposition with the canonical map ΠU → W gives the map Hk f , defined on the lower strata. Let x and y be points in the lower strata ΠU of u such that all weighted means of mixed products are non-zero, i.e. pk−1 6= 0. It is possible to choose a family of vectors Mf defined for (x, y) ∈ (ΠU ×ΠU )r{pk−1 = 0} and depending smoothly on f such that Hk f (y) − Hk f (x) = Mf · Λk (x, y) and the limit of Mf whenever x and y converge to the same point x0 ∈ ΠU is Sku f (x0 ). More precisely, for all > 0 there is a neighborhood U of f in Frö(W, Rn ) and a compact neighborhood K of x0 in ΠU and g ∈ U such that |Mg − Sku f (x0 )| < , when x, y ∈ K, pk−1 6= 0 and where Mg is constructed from g. 17 Proof: We can pull back functions by the canonical map ΠU → W . Without loss of generality we can consider f as a function ΠU → R having the form of a Hadamard expansion, see Definition 5.2. A standard proof for the mean value theorem starts by rewriting the difference f (y) − f (x) as a line integral Z ∇f · dc, f (y) − f (x) = C where C is the straight line in ΠU parameterized by c(t) = ty + (1 − t)x, 0 ≤ t ≤ 1. Introducing the scaled direction vector we may further rewrite the integral as Z 1 M (t) · Λk (x, y) dt, = 0 where M (t) is the vector field along C with coordinates 1 ∂f pj−1 ∂uji c(t) Clearly Mf = R1 0 and ∂f . pj−1 ∂ρj c(t) 1 M (t) dt depends continuously on x, y and f and f (y) − f (x) = Mf · Λk (x, y). However, Mf is undefined on pairs (x, y) where the pk−1 = 0. We will show that the scaled gradient vectors continuously extend the definition of Mf to points where x = y. More precisely, the assignment ( Mf for pk−1 6= 0 (x, y) 7→ u Sk f (y) for x = y is continuous. In the lemma below we will show that Z 1 ∂ fˆj 1 ∂f dt → (x0 ) c(t) ∂uji 0 pj−1 ∂uji Z 1 ∂ fˆj 1 ∂f dt → (x0 ) c(t) ∂ρj 0 pj−1 ∂ρj and when x and y tends to x0 . Here fˆj is the remainder in the Hadamard expansion of depth j. The limits are the coordinates of Sku f (x0 ), so the proof is complete modulo the lemma. Lemma 5.8 Let x and y be elements in a compact and convex neighborhood of x0 in ΠU . There exists a neighborhood U of f in Frö(W, R) and a constant C such that Z 1 ∂ĝ ∂g 1 j dt − (y) ≤ C|x − y| c(t) pj−1 ∂uji ∂uji 0 18 for g ∈ U. Here ĝj is the remainder in the Hadamard expansion. A similar statement holds for the partial derivative in ρj -coordinates. Proof: Differentiating the Hadamard expansion we get ∂ĝj ∂g = ρ0 · · · ρj−1 . ∂uji ∂uji Thus we may rewrite the integral as Z 1 ∂ĝj 1 dt. (tρ0 + (1 − t)r0 ) · · · (tρj−1 + (1 − t)rj−1 ) ∂uji c(t) 0 pj−1 Using integration by parts with one factor being P (t) with P 0 (t) = (tρ0 + (1 − t)r0 ) · · · (tρj−1 + (1 − t)rj−1 ) and P (0) = 0, we get P (t) ∂ĝj = pj−1 ∂uji 1 Z − 0 0 1 P (t) ∇ pj−1 ∂ĝj ∂uji · (x − y) dt. Since P (t) is increasing and defines the weighted mean of mixed products by (t) P (1) = pj−1 , we observe that pPj−1 lies in the range [0, 1]. Moreover, we can ∂ĝ choose the neighborhood U such that ∇ ∂ujij is bounded on the convex subset of the lower strata. By the mean value theorem for integrals there is a vector ~v such that we get = P (1) ∂ĝj (y) − ~v · (x − y). pj−1 ∂uji ∂ 2 ĝ where the entries of ~v have the form K ∂ξ∂ujji (z) with each K ∈ [0, 1] and z between x and y. The result follows since ~v is bounded. 6 The Whitney conditions The Whitney conditions A and B describe the transition of tangential structure between strata of stratified sets. In this section we will consider smooth maps φ : W → Rn that topologically are embeddings. When all scaled Jacobi matrices Sku φ have linearly independent column vectors, we can show that the image φ(W ) satisfies the Whitney conditions. Definition 6.1 A smooth map φ : W → Rn is called an S-immersion if all depth k coordinate systems u have scaled Jacobi matrices Sku φ(x) whose column vectors are linearly independent at any point x ∈ Bu . 19 Whenever φ : W → Rn is an S-immersion and topologically an embedding, then the image φ(W ) inherits a stratification from W . If {S} is the set of strata in W , then {φ(S)} is locally finite and satisfied the axiom of the frontier because W and φ(W ) are homeomorphic. When an S-immersion φ is restricted to a stratum S, we get an ordinary immersion of smooth manφ ifolds S − → Rn . If φ in addition is a tolological embedding, then the images φ(S) are smooth submanifolds in the Euclidean space Rn . Let us now recall the Whitney conditions, see [Whi65] or [Mat70]. Definition 6.2 (Whitney condition A) Let X and Y be strata in Rn such that X lies in the boundary of Y . Let {yi } be a sequence in Y converging to some point x ∈ X. Suppose that the tangent spaces Tyi Y converge to some τ . We say that the pair of strata X and Y satisfies Whitney condition A if the situation above always implies that Tx X ⊆ τ . Definition 6.3 (Whitney condition B) Let X and Y be strata such that X lies in the boundary of Y . Consider sequences {yi } in Y and {xi } in X such that both sequences converge to x0 ∈ X. Assume that the tangent planes Tyi Y converge to some hyperplane τ , and that the secant lines from xi to yi converge to some line `. Then ` ⊆ τ . Let us first find a suitable coordinate system for a given sequence. Lemma 6.4 Assume that a topological embedding φ : W → Rn also is an S-immersion. Consider strata X and Y of φ(W ) such that X is contained in the closure of Y . Given a sequence {yi } in Y converging to x0 ∈ X there is a depth k coordinate system u based at x0 together with a subsequence of {yi } such that each yi in the subsequence uniquely corresponds to a point in U0 × (0, ∞) × · · · × Uk and the subsequence converge to 0 in U0 × [0, ∞) × · · · × Uk . Proof: By induction on the height of W . If X and Y are the same stratum, then any depth 0 coordinate system based at x0 proves the statement. For the induction step let T be a tubular domain based at x0 . For sufficiently large i each yi lies in T and corresponds uniquely to a point in U0 × (0, ∞) × L. The projection into L forms a sequence {ŷi }, and since the link is compact we may choose a subsequence {ŷi }i∈I1 converging to some point z0 in L. By induction hypothesis there is a depth k − 1 coordinate system û in L based at z0 together with a subsequence {ŷi }i∈I2 converging to 0 in U1 × [0, ∞) × · · · × Uk . Now use T and û to construct a coordinate system u in W based at x0 . The subsequence {yi }i∈I2 converge to 0 in U0 × [0, ∞) × · · · × Uk . 20 Let us use the scaled Jacobi matrices to recognize tangent spaces of strata. We will use the following notation. Given a depth k coordinate φ system u our map φ gives a map Bu − → Rn . Over the blowup Bu = U0 × [0, ∞) × · · · × Uk × [0, ∞) × Lk we define the depth k tangent space as T k Bu = Bu × Rm where m is the dimension of U0 × [0, ∞) × · · · × Uk . By construction of the scaled Jacobi matrix we get a commutative diagram T k Bu Bu Sku φ φ / T Rn / Rn . Given a point x ∈ Bu let Txk Bu denote the depth k tangent vectors based at x. Let φ∗ Txk Bu be the image under the scaled Jacobi matrix. Geometrically consider the stratum Y ⊆ Rn parameterized under φ by the points x with ρk = 0 and other ρj 6= 0. The vector spaces φ∗ Txk Bu are the tangent spaces of Y for ρk = 0, and they extend the tangent spaces to a continuous family for vector spaces for all x ∈ Bu . We will use this to prove the Whitney conditions for φ(W ). Proposition 6.5 Assume that a topological embedding φ : W → Rn also is an S-immersion. Let Y be a stratum of φ(W ) and consider a depth k coordinate system u based at x0 such that Y corresponds to U0 × (0, ∞) × · · · × Uk . If {yi } is a sequence in U0 × (0, ∞) × · · · × Uk converging to 0 in the lower strata ΠU , then Tyi Y converge to the depth k tangent space φ∗ T0k Bu . Proof: Rescaling the Jacobi matrix does not change the column space, so Tyi Y ∼ = φ∗ Tyki Bu . We have assumed that Sku φ(0) has linearly independent column vectors. Therefore the rank of Sku φ does not fall when yi converge to 0 in ΠU . By continuity of Sku φ it follows that φ∗ Tyki Bu converge to φ∗ T0k Bu . This immediately implies that φ(W ) satisfies Whitney condition A. Corollary 6.6 Let φ : W → Rn be a topological embedding and an Simmersion. If X and Y are strata of φ(W ) such that X is contained in the boundary of Y , then they satisfy Whitney condition A. 21 Proof: Let {yi } be a sequence in Y converging to some point x0 in X. Choose a suitable depth k coordinate system by the lemma above. Then Tyi Y converge to τ = φ∗ Txk0 Bu . Since the tangent space of X at x0 is φ∗ Tx00 Bu0 , it follows that Tx0 X ⊆ τ . Hence Whitney condition A holds. Proposition 6.7 Let φ : W → Rn be a topological embedding and an Simmersion. If X and Y are strata of φ(W ) such that X lies in the boundary of Y , then they satisfies Whitney condition B. Proof: By the lemma above choose a depth k coordinate system u such that there are subsequences {yi }i∈I1 and {xi }i∈I1 where each yi ∈ U0 × (0, ∞) × · · · × Uk and xi ∈ U0 , and both sequences converge to 0. For y in ΠU we have φ(y) equal to the Hadamard expansion Hk φ(y). Applying the scaled mean value theorem, Theorem 5.7, we may write φ(yi ) − φ(xi ) = Mφ · Λk (xi , yi ). Divide by the length of φ(yi ) − φ(xi ) an let ~vi = Λk (xi ,yi ) |φ(yi )−φ(xi )| . This gives φ(yi ) − φ(xi ) = Mφ · ~vi . |φ(yi ) − φ(xi )| Here the left side is a unit vector converging to a direction vector for the line `. Since all Sku φ have linearly independent column vectors there exists a constant a > 0 and a neighborhood of matrices M around Sku φ(0) such that for all vectors ~u the inequality a|~u| ≤ |M~u| holds. For sufficiently large i all the matrices Mφ appearing above all lies in this neighborhood. Now |Mφ~vi | = 1 implies that |~vi | ≤ a1 . By compactness it is possible to choose a subsequence such that {~vi }i∈I2 converge to some vector ~v . By continuity, the limit of Mφ~vi is Sku φ(0)~v . Moreover ~v 6= 0 since 1 = |Mφ~vi | = |Sku φ(0)~v |. Proposition 6.5 says that Sku φ(0) spans τ . We have shown that Sku φ(0)~v is a direction vector for `. It follows that ` ⊆ τ . Remark 6.8 Cherenack conjectures, see [BC05, Conjecture 4.1], that the upper half cone x2 +y 2 = z 2 , z > 0 is Frölicher diffeomorphic to the quotient of S 1 × [0, ∞) by collapsing S 1 × {0} to a point. However, Bentley and Cherenack were unable to prove this. We can generalize this question and 22 ask if φ : W → Rn gives a Frölicher diffeomorphism from W onto its image whenever φ is a topological embedding and an S-immersion. Here φ(W ) is given the smooth structure of a subset in Rn . However, the generalized conjecture is certainly false. The cone C on the interval [0, π2 ] maps by polar coordinates into the first quadrant Q1 ⊆ R2 . Derivations on Q1 at (0, 0) can be identified with a 2-dimensional tangent plane. Whereas any distribution λ : C ∞ ([0, π2 ], R) → R gives rise to a distinct derivation at the base point of the cone C. Nevertheless, if W is a sharp stratified space and W 0 is a coarser smooth structure on W , then the image of φ0 : W 0 → Rn is still a Whitney stratified set whenever the pullback φ to W ia a topological embedding and an Simmersion. 7 Change of coordinates and the open set of Simmersions In this section we will study how a change of coordinates affects the scaled Jacobi matrices. It turns out that linear independence of column vectors in Sk φ does not depend on the choice of coordinate system. We use this fact to show that S-immersions of a compact sharp stratified space W form an open set in Frö(W, Rn ). Translation lets us compare coordinate systems with different basepoints, but it remains to discuss coordinate systems with the same basepoint. We will first show that the link at a point in W is unique up to diffeomorphism. Then we will relate the scaled Jacobi matrices in two different coordinate systems with the same basepoint using a change of coordinate map. Lemma 7.1 If φ : W → Rn is a map such that all max depth coordinate systemes v have scaled Jacobi matrices Slv φ with linearly independent column vectors, then φ is an S-immersion. Proof: Any u not of maximal depth may be extended to some v of maximal depth. Since all columns of Sku φ also are columns in Slv φ the result follows. Definition 7.2 Let u and v be max depth coordinate systems in W . We say that u and v have the same basepoint if • the tubular domains T0 and T00 have the same basepoint in W , and • for all j we identify the links Lj−1 and L0j−1 such that Tj and Tj0 have the same basepoint in Lj−1 ∼ = L0j−1 . 23 Let us now consider the scaled Jacobi matrix in two max depth coordinate systems with the same basepoint. Proposition 7.3 Let φ : W → Rn be a smooth map. Let u and v be max depth coordinate systems with the same basepoint. Then the column spaces are equal: Col Sku φ(0) = Col Skv φ(0) Proof: By induction on height we may assume that the statement holds for all links in W . Given max depth coordinate systems u and v we consider the change of coordinate map of the bottom tubular domains given by proposition 4.2: η : C0 × [0, δ] × L → U 0 × [0, ∞) × L0 . Use Hadamard’s lemma to write φ in both T0 and T00 . We get φ = φ0 + ρ0 φ̂1 = φ0 + r0 φ̂01 . From Proposition 4.2 we know that the quotient r̂0 = r0 ρ0 is smooth and non-negative as a function on C × [0, δ] × L. Thus φ̂1 = r̂0 · φ̂01 . Write û for the coordinate system in L consisting of the last tubular domains (T1 , . . . , Tk ) from u. By Hadamard’s lemma we may block decompose the scaled Jacobi matrix over ρ0 = 0 as û ∂φ0 u Sk φ = φ̂1 Sk−1 φ̂1 . ∂u0i Compare this to the proof of Proposition 5.4. Differentiating φ̂1 = r̂0 · φ̂01 by the product rule, we get û û û Sk−1 φ̂1 = Sk−1 r̂0 φ̂01 + r̂0 · Sk−1 φ̂01 . v̂ φ̂0 equals that of S û φ̂0 . Moreover, By induction the column space of Sk−1 k−1 1 1 ∂φ0 ∂φ0 by the chain rule the column spaces of ∂u0i and ∂v0i are equal. Thus we see that ∂φ0 0 v̂ 0 v Sk φ = φ̂ S φ̂ ∂v0i 1 k−1 1 has the same column space as Sku φ. Let us first consider the passage from a coordinate system u into the stratum where some ρs 6= 0. 24 Definition 7.4 Let u be a depth k based coordinate system. An elementary change of stratum is the following construction of a new based coordinate system v where the old tubular function ρs , for some s, becomes an ordinary coordinate. The new tubular domains are given by Tj0 = Tj for j < s, 0 ∼ T = Us × (0, ∞) × Ts+1 s Tj0 = Tj+1 and for j > s. For the new s-th local stratum we have made the identification Vs ∼ = Us × (0, ∞) × Us+1 by a translation sending 0 ∈ Vs to a point (0, ρs , 0) ∈ Us × (0, ∞) × Us+1 where ρs 6= 0. Lemma 7.5 Let φ : W → Rn be a smooth map. Let v be an elementary change of stratum of a based depth k coordinate system u. Then there is a fiberwise invertible map P such that the following diagram commutes v Sk−1 φ T k−1 Bv Bv P / T k Bu ⊆ / Bu Sku φ $ / T Rn φ / Rn . Consequently φ∗ Txk−1 Bv = φ∗ Txk Bu for all x ∈ Bv . Proof: The difference between the coordinate systems u and v is that the tubular function ρs from u corresponds to an ordinary coordinate vst in v. Otherwise all rj = ρj for j < s and rj = ρj+1 for j > s. Recall that the entries of Sku φ are 1 ∂f ρ0 · · · ρj−1 ∂uji 1 ∂f ρ0 · · · ρj−1 ∂ρj with j ≤ k and with j < k. v φ has entries Similarly Sk−1 1 ∂f r0 · · · rj−1 ∂vji 1 ∂f r0 · · · rj−1 ∂rj with j ≤ k − 1 and with j < k − 1. 25 We have for j > s 1 ∂f 1 ∂f = ρs r0 · · · rj−1 ∂vji ρ0 · · · ρj−1 ∂uji 1 ∂f 1 ∂f = ρs . r0 · · · rj−1 ∂rj ρ0 · · · ρj−1 ∂ρj and Otherwise, i.e. for j ≤ s, the entries in the scaled Jacobi matrices are equal. Consequently v Sk−1 φ = Sku φ · P, where P is a diagonal matrix with 1 on the diagonal for columns corresponding to j ≤ s and ρj on the entries corresponding to j > s. In the domain of v all ρj 6= 0, thus P is invertible. The result follows. For fixed a coordinate system v we will be interested in nearby coordinate systems of various depths. Definition 7.6 Consider a coordinate system v of depth l together with a core consisting of compact neighborhoods Cj0 ⊆ Vj and constants δj0 > 0. We say that a max depth coordinate system u is based in the core of v if there is a smooth map η : C0 × [0, δ0 ] × · · · × [0, δk ] × Ck → V0 × [0, ∞) × · · · × Vl × [0, ∞) × L0l above the identity of W , where each Cj is a compact neighborhood of 0 in Uj , and the map η sends the basepoint to η(0) inside the core given by Cj0 and δj0 for j = 0, . . . , l. Proposition 7.7 Let φ : W → Rn be an S-immersion. Given a depth l coordinate system v there exists a neighborhood U of φ in Frö(W, Rn ) and a core such that for any ψ ∈ U and any max depth coordinate system u based in the core of v the scaled Jacobi matrix Sku ψ(0) has linearly independent column vectors. Proof: By downward induction of the depth of v. For a maximal depth coordinate system v we know by continuity of the scaled Jacobi matrix defined over Frö(W, Rn ) × Bv that there exists a neighborhood U of φ and a core C00 × [0, δ00 ] × · · · × Cl0 such that Slv ψ(y) has linearly independent column vectors for all ψ ∈ U and y in the core. For the start of the induction assume that u is based in the core of the max depth coordinate system v. Then there exists a chain of elementary changes of strata w0 = v, w1 , . . ., wp such that wp has the same basepoint as u. By our previous results we have l−p l l ψ∗ T0k Bu = ψ∗ Tη(0) Bwp = · · · = ψ∗ Tη(0) Bw0 = ψ∗ Tη(0) Bv . 26 Consequently Sku ψ(0) has linearly independent column vectors for all ψ ∈ U. This establishes the start of the induction. For the inductive step consider v of depth l − 1. For each z in the link L0 of v choose a tubular domain Tz0 ⊆ L0 based at z. Let Vlz be the local stratum of Tz0 and denote the link by L̂z . For each z we get a depth l coordinate system vz given by the natural map V0 × [0, ∞) × · · · [0, ∞) × Vl−1 × [0, ∞) × Vlz × [0, ∞) × L̂z → W. By the induction hypothesis there is a neighborhood Uz of φ in Frö(W, Rn ) and a core of vz specified by C00z , . . . , Cl0z and δ00z , . . . , δl0z > 0 such that Sku ψ(0) has linearly independent column vectors for all coordinate systems u based in the core of vz . Let Pz be the image of Cl0z × [0, δl0z ] × L̂z in the link L0 . The interiors P̊z give an open covering of the compact space L0 , so we may choose a finite T subcovering indexed by z ∈ I. Intersection U = z∈I Uz gives a neighborhood of φ in Frö(W, Rn ) toT gether with a core of v specified by Cj0 = z∈I Cj0z and δj0 = minz∈I δj0z where j = 0, . . . , l − 1. Consider a max depth coordinate system u based in this core of v. We claim that u also is based in the core of some vz with z ∈ I. To see this consider the smooth map η : C0 × [0, δ0 ] × · · · × [0, δk ] × Ck → V0 × [0, ∞) × · · · × Vl−1 × [0, ∞) × L0 . The projection of η(0) to the link L0 lies in the open patch P̊z for some z ∈ I. By proposition 4.2 we may shrink the Ci -s and δi -s to get a higher lifting η : C0 × · · · × [0, δk ] × Ck → V0 × · · · × Vl−1 × [0, ∞) × Vlz × [0, ∞) × L̂z . Consequently, u is based in the core of vz . Then by the induction hypothesis the scaled Jacobi matrix Sku ψ(0) will have linearly independent column vectors for all ψ ∈ U ⊆ Uz . This proves the induction step. Thus we may conclude: Theorem 7.8 Let W be a compact sharp stratified space. The set of Simmersions in Frö(W, Rn ) is open. Proof: Let φ : W → Rn be an S-immersion. For all z in W choose a tubular domain Tz based at z. Each Tz corresponds to a coordinate system vz , so there is a neighborhood Uz of φ and a core C00z , δ00z > 0 in vz such that for any u based in the core we have a scaled Jacobi matrix Sku ψ(0) with linearly independent column vectors. The patches below C00z × [0, δ00z ] × Lz cover W and we choose a finite T subcovering indexed by z ∈ I. Then U = z∈I Uz is a neighborhood of φ, 27 and for each ψ in this neighborhood and for each coordinate system u in W , we may find another coordinate system vz such that u is based in vz . Thus Sku ψ(0) has linearly independent column vectors. 8 The open set of S-embeddings Although we know that injective S-immersions have a Whitney stratified image in Rn , we have not yet described an open subset of Frö(W, Rn ) of maps with this property. In our approach we will define S-embeddings by insisting on injectivity not only for W , but also injectivity for all links latitudinally in the limit when ρ → 0. In our approach we consider a depth k coordinate system. For the longitudinal direction we use ordinary calculus via the Hadamard expansion. The longitudinal direction corresponds to the remainder in the Hadamard expansion and is treated by induction. Definition 8.1 Let φ : W → Rn be a smooth map and u a depth k based coordinate system. Consider a point x in the lower strata ΠU . Denote by Vx the orthogonal complement of φ∗ Txk Bu . Define the radial map φ⊥ x to be the composition Lk φ̂k / Rn pr / Vx . 6 φ⊥ x Definition 8.2 Let φ : X → V be a smooth map from a Frölicher space into a vector space. We say that φ is ray injective if φ(x) 6= 0 for all x ∈ X, and whenever φ(x1 ) = λφ(x2 ) for some λ > 0, then x1 = x2 in X. The notion of an S-embedding will be defined inductively. If W is a manifold, then the S-embeddings φ : W → Rn are defined to be the ordinary embeddings of manifolds. Suppose inductively that we have specified the Sembeddings Lk → Rm for all links occuring in W . For normed vector spaces V we have the usual notion of a normalization map N : V r 0 → V given by N (~v ) = |~~vv| . Using this we specify a condition for φ on the link: Definition 8.3 A map φ : W → Rn is defined to be a longitudinal Sembedding at x ∈ W if φ is an S-immersion, and for every tubular domain T based at x the radial map φ⊥ 0 at the basepoint is ray injective and the composition with normalization is an S-embedding N φ⊥ 0 : L → V0 . We say that φ is a longitudinal S-embedding if it is a longitudinal Sembedding at all x ∈ W . 28 Now we define S-embeddings of W as follows: Definition 8.4 A smooth map φ : W → Rn is called an S-embedding if φ is an injective longditudinal S-embedding. Lemma 8.5 Let W be a compact sharp stratified space. If the set of Sembeddings in Frö(W, Rn−1 ) is open and dim W < n − 1, then the set of ray injective maps φ in Frö(W, Rn ) such that N φ is an S-embedding is also open. Proof: Let U0 be the set of maps φ in Frö(W, Rn ) such that φ(x) 6= 0 for all x ∈ W . Since W is compact it follows that U0 is open. Let V be the subset of U0 consisting of ray injective φ such that N φ are S-embeddings. Normalization induces a continuous map N∗ : U0 → Frö(W, S n−1 ) where S n−1 is the sphere of unit vectors in Rn . For ~v ∈ S n−1 let U~v be the subset of maps W → S n−1 whose image does not contain ~v . Each U~v is open, and together all U~v cover Frö(W, S n−1 ) since dim W < n − 1. By stereographic projection there are isomorphisms U~v ∼ = Frö(W, Rn−1 ). Given a ray injective map φ n in Frö(W, R ) such that N φ is an S-embedding there is a ~v such that N φ lies in the open set V~v of S-embeddings in U~v ∼ = Frö(W, Rn−1 ). The inverse image of V~v under stereographic projection and N∗ is a neighborhood of φ inside V. Lemma 8.6 Let φ : W → Rn be an S-immersion where dim W < n. Assume that the sets of S-embeddings in Frö(L, Rm ) are open for all links L occurring in W and m > dim L + 1. If φ is a londitudinal S-embedding at x and T is a tubular domain based at x, then there exists a core and a neighborhood U of φ in Frö(W, Rn ) such that for all y in the patch P below the core C × [0, δ] × L and ψ ∈ U we have that ψ is a longitudinal S-embedding at y. Proof: By downward induction on depth we will prove the following. Given a depth k coordinate system u based at x there is a neighborhood U of φ and a core such that for all y in the corresponding patch ψ is a longitudinal S-embedding at y. The claim of the lemma is the statement for k = 0. The statement holds trivially for max depth coordinate systems since they have trivial links. For the induction step consider some u based at x and with depth k. Let Vx be the orthogonal complement of φ∗ Txk Bu . Let G be the Grassmanian of dim Vx subspaces V in Rn . Consider the associated bundle Ek → G with fibers Frö(Lk , V ). Let US-imm be the open set of S-immersions in 29 Frö(W, Rn ). The construction of the radial map gives us a continuous mapping radialk : US-imm × U0 × [0, ∞) × · · · × Uk → Ek . More precisely radialk (ψ, y) = ψy⊥ . Let V be the subset of Ek consisting of ξ : Lk → V that are ray injective and have normalizations N ξ that are S-embeddings. Since ψ is an S-immersion we have dim V = n − dim(ΠU ). Then dim W ≥ dim(ΠU ) + 1 + dim Lk implies that dim V − 1 > dim Lk . By lemma 8.5 the set V is open. Let U0 × C 0 be a neighborhood of (φ, 0) inside radial−1 (V) where C 0 is a core. Then all ψ ∈ U0 are longitudinal S-embeddings at all y ∈ C 0 ⊂ U ⊂ W . In order to extend to points y outside the local stratum we will use induction. For each z ∈ Lk we may choose a tubular domain T z in Lk based at z to get a depth k + 1 coordinate system uz . Inductively, we get cores C z and neighborhoods U z of φ such that all ψ ∈ U z are longitudinal S-embeddings at y in the patch P z . There is a finite subcovering of Lk z ] × Lz } indexed by z ∈ I. Define the core C ∩ in u by z δk+1 by {Ck+1 T T × [0, z ∩ Cj = z∈I Cj and δj∩ = minz∈I δjz for j ≤ k. Similarly let U ∩ = z∈I U z define a neighborhood of φ. For all ψ in U ∩ and y in the patch below the core C ∩ , but outside the local stratum, we have that ψ is a longitudinal S-embedding at y. Now for all ψ in the intersection U 0 ∩ U ∩ we have that ψ is a longitudinal S-embedding at all y in the patch below C 0 ∩ C ∩ . Proposition 8.7 Let W be a compact sharp stratified space. Let n > dim W . Assume that the sets of S-embeddings in Frö(L, Rm ) are open for all links L occurring in W and m > dim L + 1. Then the set of longitudinal S-embeddings in Frö(W, Rn ) is open. Proof: By lemma 8.6 above we choose for each x ∈ W a tubular domain T x such that there is a neighborhood U x of φ and a core C x where all ψ ∈ U x are longitudinal S-embeddings at y in the patch. Since W is compact there is a finite subcovering of {P x } indexed by x ∈ I. Then S x x∈I U is a neighborhood of φ inside the set of longitudinal S-embeddings. Consequently the set of longitudinal S-embeddings in Frö(W, Rn ) is open. If we want to show that a longitudinal S-embedding ψ is an S-embedding, then we need to check injectivity. The next result does this locally: Proposition 8.8 Let φ : W → Rn be an S-embedding with dim W < n. Assume that the sets of S-embeddings in Frö(L, Rm ) are open for all links L occurring in W and m > dim L + 1. Given a depth k coordinate system u there is a neighborhood U of φ and a core such that for each ψ ∈ U the restriction of ψ to the corresponding patch is injective. 30 Proof: By induction on the height, we may assume that theorem 8.9 below already holds for sharp stratified spaces of lower height. Given W , we define the skeleton as the subspace W < of W where all strata of maximal dimension has been removed. The restriction induces a map Frö(W, Rn ) → Frö(W < , Rn ) and an S-embedding φ restricts to an S-embedding of W < . Consequently, by induction, there is an open neighborhood U < of maps ψ : W → Rn such that their restrictions to the skeleton are injective. In order to prove that such a map ψ ∈ U < is injective, it is enough to consider points x, y ∈ W where at least one, say y, lies in a top dimensional stratum, and show that ψ(x) = ψ(y) implies x = y. Internally in this proof we proceed by reverse induction on the depth k of the coordinate system. If u has maximal depth, then we may use the scaled mean value theorem 5.7. For a sufficiently small core and a neighborhood U ⊆ U < of φ there are matrices Mψ such that ψ(y) − ψ(x) = Mψ · Λk (x, y) for all pairs x, y in the core where the weighted mean of mixed products pk−1 6= 0. Moreover, since φ is an S-immersion, we may assume that all Mψ are injective. If ψ(x) = ψ(y) where y lies in a top dimensional stratum, then pk−1 6= 0. It follows that Λj (x, y) = 0, and consequently x = y. Hence all ψ in some neighborhood U are injective on the patch corresponding to the core chosen above. To prove the inductive step we will establish several inequalities. Let us state each such inequality carefully, and then prove it. Let pr t : Rn → Rn be the orthogonal projection onto φ∗ T0k Bu , i.e. the subspace of Rn spanned the scaled Jacobi matrix Sku φ(0). Let Hkt ψ = pr t Hk ψ be the tangential projection of the Hadamard expansion. As usual V0 will denote the ortogonal complement of φ∗ T0k Bu . Let pr n be the projection onto V0 , and let Hk⊥ ψ = pr n Hk ψ. In this proof x and y will be points in the blowup of u, Bu = U0 × [0, ∞) × · · · × [0, ∞) × Uk × [0, ∞) × Lk . The coordinates corresponding to tubular functions are r0 , . . . , rk for x and ρ0 , . . . , ρk for y. The weighted means of mixed products, p0 , . . . , pk , are then given by the formula in Definition 5.5. Let r and ρ be the products r = r0 · · · rk−1 rk and ρ = ρ0 · · · ρk−1 ρk . Since y lies in a top dimensional stratum, both ρ 6= 0 and pk−1 6= 0. We will omit the projection Bu → ΠU from our notation and write Hk ψ(y) even though the domain of the Hadamard expansion is the lower strata ΠU . 31 Claim 1: There exist a constant K1 > 0, a core and a neighborhood U of φ such that for all ψ in U and points x, y in the core with pk−1 6= 0, we have t H ψ(x) − H t ψ(y) ≥ K1 |Λk (x, y)| . k k Proof of claim 1: The function given by f (~v ) = |pr t Sku φ(0) ~v | has a minimum value c on the set of unit vectors ~v . We can take c > 0 since pr t Sku φ(0) is injective. Since the limit of Mψ is Sku φ(0) as ψ → φ and x, y → 0, there is a core and a neighborhood U of φ such that |pr t Mφ~v | ≥ Let ~v = Λk (x,y) |Λk (x,y)| c 2 for all x, y, ψ, ~v with pk−1 6= 0. and K1 = 2c . Then |pr t Mφ Λk (x, y)| ≥ K1 |Λk (x, y)| . The inequality in the claim now follows by the scaled mean value theorem. Claim 2: There exist a constant K2 > 0, a core and a neighborhood U of φ such that for all ψ in U and points x, y in the core with pk−1 6= 0, we have p ρ2 + r2 ≥ K2 pr t (ρψ̂(y) − rψ̂(x)) . Proof of claim 2: Given an upper bound R on the norm of two vectors ~v1 and ~v2 , there is a constant c such that c ≥ |cos θ ~v1 − sin θ ~v2 | for all θ ∈ 0, π2 . Let R be twice the maximum of |pr t φ̂(0, . . . , 0, z)| for z in L. Then there are a core and a neighborhood U of φ such that for ψ ∈ U R ≥ |pr t ψ̂(x)| for all x in the core. Suppose p r · ρ 6= 0. Take p ~v1 = ψ̂(y), ~v2 = ψ̂(x) and let θ be given by ρ = ρ2 + r2 cos θ, r = ρ2 + r2 sin θ. Furthermore set K2 = 1c . Then the inequality p ρ2 + r2 ≥ K2 pr t (ρψ̂(y) − rψ̂(x)) holds by a rearranging of the inequality above. In the case r · ρ = 0 the inequality is trivially true. 32 Claim 3: Let {T i } be a finite set of tubular domains in Lk together with cores C i × [0, δ i ] × Li whose image cover Lk . Using that φ is a longitudinal S-embedding there exist a constant K3 , a core of u and a neighborhood U of φ such that for all ψ ∈ U and points x, y in the core of u, we have there is an index i such that both x and y lie in the core C0 × [0, δ0 ] × · · · × Ck × [0, δk ] × C i × [0, δ i ] × Li or the following inequality holds, p pr n (ρψ̂(y) − rψ̂(x)) ≥ K3 ρ2 + r2 . Proof of claim 3: For any pair of vectors ~v1 and ~v2 let β(~v1 , ~v2 ) be defined as the minimum of |cos θ~v1 − sin θ~v2 | for θ ∈ 0, π2 . Let D be the open neighborhood of the diagonal in Lk × Lk consisting of pairs (z1 , z2 ) such that there exists an inxed i with both z1 and z2 interior points of the core C i × [0, δ i ] × Li of T i . Since φ is longitudinal S-embedding the radial map at the basepoin φ⊥ 0 is ray injective, and there exists a number δ > 0 such that β(pr n φ̂(x), pr n φ̂(y)) ≥ δ for (x, y) ∈ (L × L) r D. Choose a neighborhood U of φ and a core of u such that δ β(pr n ψ̂(x), pr n ψ̂(y)) ≥ 2 for all ψ ∈ U and x, y inside the core such that the longitudinal projection of (x, y) onto Lk × Lk does not lie in D. With K3 = 2δ and θ as in claim 2 this implies the inequality of claim 3 for (x, y) not above D, but if (x, y) lies above D, then there is an index i such that both x, y ∈ C i × [0, δ i ] × Li . This proves the claim. Claim 4: Given any constant > 0 there exist a core and a neighborhood U of φ such that for all ψ in U and points x, y in the core with pk−1 6= 0, we have |Λk (x, y)| ≥ Hk⊥ ψ(x) − Hk⊥ ψ(y) . Proof of claim 4: Observe that the matrix pr n Mψ converges to 0 as ψ goes to φ and x, y → 0. Given > 0 we may choose a core and a neighborhood U such that for all ψ ∈ U, x, y in the core with pk−1 6= 0 and all unit vectors ~u we have ≥ |pr n Mψ · ~u| . The inequality in the claim follows by the scaled mean value theorem by Λk (x,y) setting ~u = |Λ . k (x,y)| 33 Proof of proposition: For each z ∈ Lk choose a tubular domain T z based at z. This gives depth k + 1 coordinate systems uz given by (T0 , . . . , Tk , T z ). By induction, there are a core C z of uz and a neighborhood U z of φ inside U < such that each ψ in U z is injective on the patch P z below the core C z . Since Lk is compact, we may choose finitely many {T i } such that the cores i i i Ck+1 ×[0, T δi k+1 ]×L cover Lki. Define a core C of u by taking the intersection Cj = Cj and δj = mini δj . Likewise define the neighborhood U0 as the intersection of the U i from the finite covering. By choosing the core and neighborhood sufficiently small, we get constants K1 , K2 and K3 such that the inequalities of claim 1, 2 and 3 are satisfied. Choose some < K1 K2 K3 , and shrink the core and neighborhood further such that the inequality in claim 4 also holds. Assume that ψ(x) = ψ(y) for some ψ ∈ U and points x, y in the core with pk−1 6= 0. We will show that x and y are identified in W . By the Hadamard expansion Hk ψ(x) − Hk ψ(y) = ρ0 · · · ρk ψ̂(y) − r0 · · · rk ψ̂(x). The four inequalities now imply that K1 K2 K3 |Λk (x, y)| ≤ K2 K3 Hkt ψ(x) − Hkt ψ(y) = K2 K3 pr t (ρψ̂(y) − rψ̂(x)) p ≤ K3 ρ2 + r2 ≤ pr n (ρψ̂(y) − rψ̂(x)) = Hk⊥ ψ(x) − Hk⊥ ψ(y) ≤ |Λk (x, y)| . p The only possible solution is |Λk (x, y)| = 0 and ρ2 + r2 = 0, but this contradicts the assumption that y lies in the top dimensional stratum. Consequently the inequality of claim 3 does not hold. Therefore x and y lie in the core of one of the deeper coordinate systems ui . By induction ψ is injective on this deeper patch P i . Hence x = y. Thus all ψ in U are injective on the patch P of u. Theorem 8.9 Let W be a compact sharp stratified space and n > dim W . Then the set of S-embeddings is open in Frö(W, Rn ). Proof: By induction on height of W . Let φ : W → Rn be an S-embedding. There is a neighborhood U1 of longitudinal S-embeddings by Proposition 8.7. Cover W by tubular domains T x . Each T x is a depth 0 coordinate system, 34 and by Proposition 8.8 there is a core C x and a neighborhood U x such that for all ψ ∈ U x we have injectivity of ψ restricted to the patch P x below C x . By compactness of W there is a finite subcovering {P i } and a Lebesgue number δ > 0 such that if |φ(x) − φ(y)| < δ then x, y ∈ P i for some patch from the finite subcoveing. Let U0 be the neighborhood of φ consisting of all maps ψ such that |ψ(y) − φ(y)| < T2δ for all y ∈ W . We claim that all ψ in U0 ∩ U1 ∩ U i are S-embeddings. Since ψ ∈ U1 the map is a longitudinal S-embedding. Assume that ψ(x) = ψ(y). By the triangle inequality |φ(x) − φ(y)| ≤ |φ(x) − ψ(x)| + |ψ(y) − φ(y)| < δ δ + =δ 2 2 it follows that x, y ∈ P i for some patch from the finite subcovering. Then x = y since ψ ∈ U i is injective on P i . Consequently all such ψ are injective longitudinal S-embeddings. 9 Existence of S-embeddings In this section we will prove that S-embeddings exist into Rn when n is sufficiently large. Furthermore the set of S-embeddings is dense among all smooth maps. We will follow the approach found in [Hir76]. In order to prove these statements, we need a direct way of identifying S-embeddings. We begin by noting a longitudinal version of the chain rule: Lemma 9.1 (The longitudinal chain rule) Let φ : W → Rn be smooth and let g : Rn → R be a function. Consider the composition f = g ◦ φ. For a tubular domain T we apply Hadamards lemma and find fˆ1 = Dgφ (x) · φ̂1 . ρ0 =0 0 ρ0 =0 Proof: Differentiating the equality f = f0 + ρ0 fˆ1 with respect to ρ0 and then substituting ρ0 = 0 we get ∂f ˆ1 = f . ρ =0 ρ0 =0 ∂ρ0 0 By the ordinary chain rule for g ◦ φ = g ◦ (φ0 + ρ0 φ̂1 ) we get ! ∂ ∂ φ̂1 (g ◦ φ) = Dgφ(x) · φ̂1 + ρ0 . ∂ρ0 ∂ρ0 By substituting ρ0 = 0 the result follows. 35 Proposition 9.2 Let φ : W → Rn be an S-immersion. Then the following are equivalent: i) The map φ is a longitudinal S-embedding. k ii) For all depth k coordinate systems u the radial maps φ⊥ 0 : Lk → V0 are ray injective at the basepoint. Proof: Let us prove that i) and ii) are equivalent by induction on the height of W . This is tautologically true for manifolds. For the inductive step consider a tubular domain T based at x in W . Consider the corresponding radial map φ⊥ 0 : L → V0 . Since T is a depth 0 coordinate system, we observe that ray injectivity of φ⊥ 0, ⊥ as in statement ii), is equivalent to injectivity of the normalization N φ0 , as in statement i). Let ψ = N φ⊥ 0 : L → V0 . By the induction hypothesis, ψ being a longitudinal S-embedding is equivalent to the following statement: For all depth k − 1 coordinate systems û of L the radial map ψ0⊥ : L̂k−1 → V̂0k−1 at the basepoint of û is ray injective. Here V̂0 is the orthogonal complement of ψ∗ T0k−1 Bû . Given such a depth k − 1 coordinate system û in L we construct the coordinate system u in W by T0 = T and Tj = T̂j−1 for j > 0. Then Lj = L̂j−1 . Now there are several subspaces of Rn to consider. From the tubular domain T there is the normal plane of the local stratum U at the basepoint, which we denote V0 . The scaled Jacobi matrix Sku φ(0) spans φ∗ T0k Bu , and V0k is its orthogonal complement in Rn . Note that V0k ⊆ V0 . Using ψ and û defined on the link L we get the subspace ψ∗ T0k−1 Bû in V0 û ψ(0). The complement in V is called V̂ k−1 . spanned by Sk−1 0 0 We now want to compare the radial maps k φ⊥ 0 : Lk → V 0 and ψ0⊥ : L̂k−1 → V̂0k−1 . We get φ⊥ 0 from the expansion φ = φ0 + ρ0 φ̂1 = φ0 + ρ0 φ1 + . . . + ρ0 · · · ρk−1 φ̂k p by the formula φ⊥ 0 = pr V0 φ̂k . Let N be the composition of orthogonal projection Rn → V0 and the normalization map N (~v ) = |~~vv| in V0 . We observe that N p is defined for all ~v ∈ Rn not in the tangent plane of φ0 : U → Rn at the basepoint. With this notation we have ψ = N p ◦ (φ1 + ρ1 φ2 + . . . + ρ1 · · · ρk−1 φ̂k ). 36 By the chain rule we can compute the scaled Jacobi matrix û û Sk−1 ψ(0) = DNφp1 (0) Sk−1 φ̂1 (0). Furthermore, we get by the longitudinal chain rule that the map ψ0⊥ : L̂k−1 → V̂0k−1 is given as ψ0⊥ = pr V̂ k−1 DNφp1 (0) φ̂k . 0 Since φ is an S-immersion it follows that the vector φ1 (0), which is one of the columns in Sku φ, is orthogonal to V0k . By computation of the Jacobi matrix of the normalization we have pr V̂ k−1 DNφp1 (0) = 0 Hence ψ0⊥ = 1 pr k . |φ1 (0)| V0 1 incl ◦φ⊥ 0, |φ1 (0)| where incl is the inclusion of V0k into V̂0k−1 . By this formula it follows that ψ0⊥ is ray injective if and only if φ⊥ 0 is ray injective. This completes the inductive step. Lemma 9.3 If φ : W → Rn is an S-embedding and f : W → R a smooth function, then (φ, f ) : W → Rn+1 is also an S-embedding. Proof: Let u be a depth k coordinate system. For the scaled Jacobi matrices we have u Sk φ u Sk (φ, f ) = . Sku f Since the column vectors of Sku φ are linearly independent, the larger matrix also has this property. Moreover we have Vk = Vk0 ∩ Rn , where Vk and Vk0 are the orthogonal complements of the column spaces of Sku φ in Rn and Sku (φ, f ) in Rn+1 respectively. Consequently, pr Vk pr Rn = pr Vk pr V 0 . This implies k that ⊥ ˆ pr Vk (φ, f )⊥ k = pr Vk pr V 0 (φ̂k , fk ) = pr Vk φ̂k = φk . k Suppose that y1 , y2 in Lk satisfies ⊥ (φ, f )⊥ k (y1 ) = λ(φ, f )k (y2 ) for some λ > 0. Apply pr Vk to transform this equation into ⊥ ⊥ ⊥ φ⊥ k (y1 ) = pr Vk (φ, f )k (y1 ) = λpr Vk (φ, f )k (y2 ) = λφk (y2 ). 37 ⊥ Then y1 = y2 since φ⊥ k is ray-injective. It follows that (φ, f )k also is ray injective. Since u was general, Proposition 9.2 implies that (φ, f ) is an S-embedding. Lemma 9.4 A compact sharp stratified space W can be S-embedded in RN for some large N . Proof: By induction on the height of W . Manifolds can be embedded by the well-known Whitney embedding theorem, see [Hir76, Theorem 1.3.5]. For the inductive step first consider a tubular domain T with link L and local stratum U ⊆ Rk . By induction choose an S-embedding ψ : L → Rn . Define φ : T → Rk+n+1 by φ = φ0 + ρ0 φ̂1 , where φ0 is the inclusion U ⊆ Rk ,→ Rk+n+1 and φ̂1 is the composition ψ inv. stereo. proj. L− → Rn −−−−−−−−−−→ Rn+1 ,→ Rk+n+1 of ψ with the inverse stereographical projection and the inclusion of Rn+1 as the last coordinates in Rk+n+1 . Clearly this map φ is a longitudinal S-embedding and it is injective, thus φ is an S-embedding. For a compact sharp stratified space W choose a covering by tubular domains {T i }, i = 1, 2, . . . , K, together with a partition of unity {gi : W → R} where gi has support in T i . For each i there exists an S-embedding φi : T i → Rni . Define ψ i : W → Rni +1 by ψ i (x) = (gi (x), gi (x) · φi (x)). Clearly each ψ i is a longitudinal S-embedding at points where gi (x) > 0. Then by the lemma above we get that (ψ 1 , ψ 2 , . . . , ψ K ) : W → RN , where N = n1 + n2 + . . . + nK + K, is everywhere a longitudinal S-embedding. However it is easily verified that this map is also injective. This constructs an S-embedding W → RN for some large N . Given a unit vector ~u in Rn+1 we define the projection p~u : Rn+1 → Rn as the composition of orthogonal projection onto the normal space of ~u and thereafter the projection down to Rn ⊆ Rn+1 . The last projection is bijective when ~u ∈ / Rn . 38 Proposition 9.5 Let W be a compact sharp stratified space of dimension m. Let φ : W → Rn+1 be an S-embedding. If 2m + 1 ≤ n, then the set of ~u ∈ RP n such that p~u φ : W → Rn is not an S-embedding, has measure 0. Proof: It is well known that the image of an m-manifold in an n-manifold has measure 0 when m < n. We will construct finitely many subsets of RP n having measure 0 and such that for all ~u not in the union the composition p~u φ is an S-embedding. First of all we want ~u ∈ / RP n−1 . Suppose X 6= Y are strata of W . Let eX,Y : X × Y → RP n be the smooth map sending (x, y) to the line with direction vector φ(x) − φ(y). Since dim X + dim Y ≤ 2m < n the image EX,Y ⊆ RP n has measure 0. Observe that if ~u ∈ / EX,Y ∪ RP n−1 then there is no intersection p~u φ(x) = p~u φ(y), with x ∈ X and y ∈ Y . Similarly for X a stratum consider eX : X ×X r∆ → RP n given by sending (x1 , x2 ), with x1 6= x2 , to the line with direction vector φ(x1 ) − φ(x2 ). The image EX ⊆ RP n has again measure 0. Define E to be the union [ [ E= EX,Y ∪ EX . X X6=Y Now whenever ~u ∈ / E ∪ RP n−1 , then p~u φ : W → Rn is injective. Next consider a max depth coordinate system u. Using the scaled Jacobi matrix we define a map fu : ΠU × RP m−1 → RP n sending a point u ∈ ΠU and a direction vector ~v ∈ RP m−1 to the line with direction vector S u φ(u) · ~v . The source is a manifold (with boundary) of dimension 2m − 1. When 2m − 1 < n the image Fu of this map is a subset in RP n of measure 0. Cover W by finitely many max depth coordinate systems, and let F be the union of the corresponding Fu . Observe that the composition p~u φ is an S-immersion whenever ~u ∈ / F ∪ RP n−1 . At last consider a depth k coordinate system u with link Lk . We want to construct subsets such that the composition p~u φ has ray injective radial maps for all ~u not in these subsets. Let X 6= Y be strata in the link Lk . We will define a smooth gu,X,Y : ΠU × (0, ∞) × X × Y × Rl → RP n , where l is the number of columns in Sku φ. This map sends (u, λ, z1 , z2 , ~v ) to the line with direction vector φ̂(u, 0, z1 ) − λφ̂(u, 0, z2 ) + S u φ(u) · ~v . 39 To check that gu,X,Y is well defined, suppose that φ̂(u, 0, z1 ) − λφ̂(u, 0, z2 ) + S u φ(u) · ~v = 0. Then φ̂(u, 0, z1 ) − λφ̂(u, 0, z2 ) ∈ Vuk , i.e. φ⊥ (z1 ) = λφ⊥ (z2 ). But this is impossible because the radial map φ⊥ is ray injective. Let Gu,X,Y be the image of gu,X,Y in RP n . Since dim W = m it follows that l + 1 + dim X ≤ m, and similar for dim Y . Hence dim ΠU × (0, ∞) × X × Y × Rl ≤ l + 1 + dim X + dim Y + l < 2m. Consequently, Gu,X,Y has measure 0 in RP n . Similarly, we may define a map gu,X : ΠU × (0, ∞) × (X × X r ∆) × Rl → RP n , using the same formula as above. Again the image Gu,X is a measure 0 subset of RP n . Define Gu to be the union [ [ Gu = Gu,X,Y ∪ Gu,X . X X6=Y where X, Y run through all strata of Lk . Let ~u be a direction vector for some line in RP n r RP n−1 and suppose that (p~u φ̂)⊥ is not ray injective at some u ∈ ΠU . Then there are points z1 6= z2 ∈ Lk and a positive number λ such that (p~u φ̂)⊥ (z1 ) = λ(p~u φ̂)⊥ (z2 ). Hence for some ~v ∈ Rl we have p~u φ̂(u, 0, z1 ) − λp~u φ̂(u, 0, z2 ) + (Sku (p~u φ)(u)) · ~v = 0. This implies that there exists a scalar r such that φ̂(u, 0, z1 ) − λφ̂(u, 0, z2 ) + Sku φ(u) · ~v = r~u. Therefore ~u ∈ Gu,X,Y where X and Y are the strata of z1 and z1 respectively, or ~u ∈ Gu,X if z1 , z2 lie in the same stratum. Covering the compact sharp stratified space S W by finitely many coordinate systems, we construct a set G = Gu of measure 0 in RP n . We have shown above that all radial maps of p~u φ are ray injective when ~u ∈ / G ∪ RP n−1 . Summarizing, we see that p~u φ is an S-embedding when ~u lies outside the measure 0 set E ∪ F ∪ G ∪ RP n−1 . 40 Theorem 9.6 Let W be a compact sharp stratified space of dimension m. The set of S-embeddings is dense in Frö(W, Rn ) provided that 2m + 1 ≤ n. Proof: Let Un ⊆ RP n be the direction vectors represented by ~u ∈ / Rn−1 . Composing with the projections p~u we get a continuous map p : Un × Frö(W, Rn+1 ) → Frö(W, Rn ). Let φ0 : W → Rn be an arbitrary smooth map. Consider an open neighborhood U of φ0 in Frö(W, Rn ). Choose an S-embedding ψ : W → RN . Then (φ0 , ψ) : W → Rn+N is also an S-embedding. By iterated composition of the maps p above we get a map Un × · · · × Un+N −1 × Frö(W, Rn+N ) → Frö(W, Rn ). Observe that (~en+1 , . . . , ~en+N , (φ0 , ψ)) maps to φ0 . By continuity there are neighborhoods Vj of ~ej+1 in Uj such that Vn × · · · × Vn+N −1 × {(φ0 , ψ)} has image in U. 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