A Whitney embedding theorem for a class of

stratified Frolicher spaces

Tore August Kro, tore.a.kro@hiof.no

March 7, 2016

Abstract

Classically Whitney described stratified sets as subsets of Rn splitting as adisjoint union of submanifolds satisfying conditions A and B. To take a moremodern approach to stratified spaces, choose some category of generalizedsmooth spaces, define a list of allowed types of neighborhoods, and thendeclare W to be stratified if every point in W has a neighborhood diffeo-morphic one of the allowed types. In this paper we employ Frolicher spaces,which is a complete, cocomplete and Cartesian closed category. We allowneighborhoods T = U ×CL being the product of an open subset U of someEuclidean vector space and a cone CL. Note that smooth function spaces,for example Fro(W,Rn), appear automatically as categorical constructions.The aim of this paper is to generalize the Whitney embedding theorem toour class of stratified spaces. We define S-embeddings φ : W → Rn andshow that for compact W of dimension m, the set of S-embeddings is openand dense in the mapping space Fro(W,Rn) when n > 2m. Moreover, theimages φ(W ) of S-embeddings are Whitney stratified sets in the classicalsense.

1 Introduction

The field of differential topology has produced fascinating results aboutmanifolds. Among these are the sphere eversion [Sma58, FM80], smoothstructures on the 7-sphere, [Mil56, Wal99] and surveyed in [Luc02], andthe calculation of the mapping class group [MW07, GTMW09]. Stratifiedspaces generalize the concept of manifolds. Classically, for stratified subsetsof manifolds Whitney introduced in [Whi65] his conditions A and B on howtangent planes behave when approaching a lower stratum. Topologicallya stratification of X is a cover consisting of pairwise disjoint manifolds ofvarying dimension. They are usually required to satisfy locally finitenessand the axiom of the frontier, basically describing how the boundary of

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one stratum is the union of lower dimensional strata. These axioms can befound in [Tho69, Tho71, Mat70]. There are several other axioms for strati-fied subsets of smooth manifolds, both stronger and weaker than Whitney’scondition B. These regularity conditions are surveyed by Trotman in [Tro07].

There are several classes of topological stratified sets. Siebenmann in-troduced in [Sie72] locally conelike TOP stratified sets. His definition isanalogous to the sharp stratified spaces introduced in this paper, the dif-ference is mainly the category in which the construction takes place. LaterQuinn introduced in [Qui88] manifold homotopically stratified sets in orderto give a setting for the study of purely topologically stratified phenomena.Hughes and Weinberger survey in [HW01] these topologically stratified setsthrough applications of intersection homology and surgery theory.

In order to do differential topology for stratified spaces one would liketo define the notion of a smooth structure on a topologically stratified setand generalize tools such as transversality, Morse theory, the Whitney em-bedding theorem and many other results. Moreover, there should be newresults, such as Thom’s first and second isotopy lemmas [Mat70]. Defin-ing the smooth structure on a stratified space has its difficulties. Matherintroduce abstract pre-stratified sets by giving data on how tubular neighbor-hoods of nearby strata relates. This amount of data leads to a technicallydemanding theory, and function spaces of smooth maps are not immedi-ately given. Later Mathias Kreck [Kre10] introduced stratifolds by usingSikorski’s differential spaces [Sik72] to keep track of the smooth structure.Unfortunately, for any stratifold S, which is not a manifold, there exists nosmooth injective map S ↪→ Rn for any finite n. In particular, for the halfopen 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 devel-oped. Boman shows in [Bom67] that a function f : Rn → R is smooth ifand only if the composition f ◦c is smooth for all smooth curves c : R→ Rn.Frolicher uses this property to define a category of smooth spaces [Fro82].This category is now known as Frolicher spaces. A result due to Law-vere, Schanuel and Zame [LSZ81] implies that Frolicher spaces is Cartesianclosed. Already Frolicher notes that his category includes objects with sin-gularities, but together with Kriegl and Michor the theory is developed inthe direction of infinite dimensional spaces [KM97, FK88]. In this direc-tion Cap [Cap93] developes in his thesis a K-theory for convenient algebras.In [Che00] Cherenack begins to consider the finite dimensional aspects ofFrolicher spaces. Cherenack asks in [BC05] if the Frolicher structure on thecone S1× [0,∞) /S1× 0, constructed as a pushout, is equal to the subspacestructure of a cone embedded in R3. Another interesting question is thetopology on the Frolicher space of maps with compact support, see [Che01].Dugmore and Ntumba [DN07] does homotopy theory in Frolicher spacesby making use of flattened and weakly flattended unit intervals. While

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Batubenge and Tshilombo [BT09, BT14] study topologies on final and ini-tial objects, products and coproducts. A nice collection of finite dimensionalexamples appear in [HKM14]. The development of Frolicher spaces has re-cently been surveyed by Batubenge [Bat15]. Moreover a comparative studyby Stacey [Sta11] relates Frolicher spaces to other alternative categories ofgeneralized smooth spaces.

In this paper we define our class of stratified spaces by specifying whattypes of neighborhoods that are allowed. A sharp stratified space W isbasically a Frolicher space where each point has a neighborhood T , called atubular domain. These tubular domains are products T = U × CL, whereCL is a cone and U an open subset of some Euclidean space. The mainresult is a Whitney embedding theorem for compact sharp stratified spaces.It is not hard to generalize this local approach and define larger classes ofstratified spaces. Prominent examples of types of neighborhoods that couldbe included are orbit spaces V/G, where G is a compact Lie group actingsmoothly on a finite dimensional vector space V . However, the aim in thispaper 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 def-inition of a Frolicher space. We construct sharp stratified spaces in section 3.Our main technical tool is Hadamards lemma 3.5. Moreover, we prove thatsharp stratified spaces are topologically stratified. In section 4 we discussthe topology of the mapping spaces. Noting that they are Frechet spaces,we get an explicit description of the open sets. In section 5 we introducecoordinate systems, the scaled Jacobi matrix, and prove a scaled mean valuetheorem. In section 6 we discuss the classical Whitney conditions and showthat the hold for reasonable images of sharp stratified spaces into Euclideanspace. In section 7 we show that the set of S-immersions is open in the map-ping space. Furthermore, we show in section 8 that the set of S-embeddingsalso is open. There are plentiful of S-embeddings. In section 9 we showthat the set of S-embeddings is dense if the dimension of the codomain issufficiently large. Our main result is:

Theorem 1.1 Given a compact sharp stratified set W of dimension m. Ifn > 2m, then there is an open and dense set of mappings φ : W → Rn inthe function space Fro(W,Rn) such that φ is injective and the image φ(W )is a Whitney stratified set.

2 About Frolicher Spaces

This section is a concise summary of Frolicher spaces. We recall the defini-tion of the category Fro, and review how it is a complete, cocomplete andCartesian closed category. Boman’s theorem shows that smooth manifolds

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are contained in Fro as a full subcategory. Furthermore, we discuss topolo-gies on a Frolicher space. The half open interval is of particular interest asit is used in the construction of tubular domains.

Definition 2.1 A Frolicher space X is a triple (X, CX ,FX) where X is aset, 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 fc ∈ C∞(R,R) for all f ∈ FX ,and

f : X → R lies in FX if and only if fc ∈ C∞(R,R) for all c ∈ CX .

The pair (CX ,FX) is the smooth structure on X. We call CX and FXthe smooth curves and smooth functions respectively. A map φ : X → Ybetween the underlying sets of two Frolicher spaces is called smooth if foreach f ∈ FY , the pullback fφ lies in FX . Let Fro denote the category ofFrolicher 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) sayingthat for all c ∈ CX we have φc ∈ CY , and (3) saying that for all c ∈ CX andf ∈ FY the composition fφc lies in C∞(R,R).

The real numbers R form a Frolicher space by declaring CR = FR =C∞(R,R). The n-dimensional Euclidean space Rn is a Frolicher space wherethe 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 thiscase, namely that f : Rn → R is smooth if the composed function fc lies inC∞(R,R) for every smooth curve c in Rn.

The category Fro is both complete and cocomplete, see [FK88, Corol-lary 1.1.5]. A smooth structure (CX ,FX) on X is finer than another smoothstructure (C′X ,F ′X) on the same set if CX ⊆ C′X (or equivalently F ′X ⊆ FX).Given any set C0 ⊆ XR there is a finest smooth structure on X such thatC0 ⊆ CX ; we call it the smooth structure generated by C0 and we have

FX = {f : X → R | fc ∈ C∞(R,R) for all c ∈ C0}.

Similarly, given any set F0 ⊆ RX there is a coarsest smooth structure onX generated by F0, i.e. F0 ⊆ FX . Consequently, the forgetful functorU : Fro → Set has both a left and a right adjoint. It follows that U preservesboth limits and colimits, and the underlying set of a (co)limit is the (co)limitof the underlying sets.

To prove that Fro is Cartesian closed one needs to choose a smooth struc-ture on C∞(R,R) satisfying certain conditions, see [Fro82] and [FK88, The-orem 1.1.7]. The condition is verified in [FK88, Theorem 1.4.3]. The curvesc in a smooth structure on C∞(R,R) can be defined precisely as the maps

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c : R → C∞(R,R) such that the adjoint c : R× R → R is smooth. Equiva-lently, c is smooth if and only if it is a smooth curve c : R→ C∞(R,R) whenC∞(R,R) is considered as the Frechet space with the topology of uniformconvergence of each derivative on compact sets, see [KM97, Theorem 3.2].More generally, the smooth structure on the internal hom Fro(X,Y ) is de-fined to have smooth curves given by

CFro(X,Y ) = {c : R→ Fro(X,Y ) | the adjoint c : R×X → Y is smooth}.

As a direct consequence of being Cartesian closed, it follows that takingcross product with a fixed Frolicher space commutes with taking colimits.

In general there may be several interesting topologies on a Frolicherspace X. As noted by Cap in his thesis [Cap93, Paragraph 1.9], there aretwo extremes. We always want R to have the standard topology. UsingStacey’s names for these topologies, see [Sta13, Definition 2.1], we definethe functional topology on X as the coarsest topology such that all smoothfunctions f : X → R are continuous. The curved topology on X is the finesttopology such that all smooth curves c : R→ X are continuous. The curvedtopology is called the c∞-topology in [KM97, Definition 2.12]. For someX the two extreme topologies are the same, and in that case X is usuallycalled balanced following Cap’s terminology, or smoothly regular by Stacey.All manifolds are balanced. When X is a function space, there are in generalseveral different topologies, see [KM97, Example 4.8].

A Frolicher space X is called Hausdorff if the functional topology, hencealso the curved topology, on X is Hausdorff, i.e. if for any pair of distinctpoints p1, p2 in X there is a smooth function f ∈ FX with f(p1) 6= f(p2).Similarly, we define a Frolicher space to be locally compact, compact, para-compact, or to have a countable basis for its topology, etc. if this holds forits functional topology.

Proposition 2.2 A paracompact Frolicher space admits smooth partitionsof unity subordinate to any open cover.

Proof: Let {Uα} be an open cover of a paracompact Frolicher space X. Bydefinition of the functional topology on X we can cover X with smooth non-negative functions fβ such that the support of each fβ is contained in someUα. Pass to a locally finite refinement of the family of functions {fβ}. Foreach Uα construct a new function by taking the sum of all fβ supported inUα. Normalization gives a smooth partition of unity subordinate to {Uα}. �

On the half open interval [0,∞) we take the Frolicher subspace structurecorresponding to the inclusion [0,∞) ⊆ R, i.e. C[0,∞) are all curves c : R→[0,∞) such that the composition

R c−→ [0,∞)⊆−→ R

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is smooth. Bentley and Cherenack [BC05, Proposition 3.1] characterize thesmooth functions F[0,∞) as functions f : [0,∞) → R which are smoothat all interior points of [0,∞) and right smooth at 0. This follows fromthe more general fact that Frolicher smooth functions on a codimension 0smooth submanifold M with boundary of a manifold E extends to a smoothfunction 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 ofFrolicher spaces by identifying t ∈ R with −t ∈ R. Bentley and Cherenackprove that the map

η : R/± → [0,∞) given by η(x) = x2

is a diffeomorphism of Frolicher spaces.There are several other smooth structures on the half open interval.

Dugmore and Ntumba [DN07] define, for the purpose of doing homotopytheory, a flattened and a weakly flattened unit interval. However, we willnot use these structures in this paper.

3 Construction

Being a stratified space should be a local property. One approach to theirconstruction is thus given by specifying a collection of allowed neighborhoodtypes, and defining a stratified space as a Frolicher space locally diffeomor-phic to one of these neighborhoods. Actually this construction can also pro-ceed in several steps, starting with the collection of all finitely dimensionalEuclidean spaces. The sharp stratified spaces will be constructed below byinductively considering certain pushouts over the half open interval as al-lowed neighborhoods. Certainly this could be generalized to other types ofstratified neighborhoods, but the aim of this article is not to give the mostgeneral construction of stratified space imaginable, but rather to reproduceWhitneys embedding theorem.

We will call our allowed neighborhoods tubular domains. They are de-fined recursively, but for the initial step the space denoted by L below isjust a compact manifold.

Definition 3.1 A tubular domain T of height h is the pushout in Frolicherspaces given in the diagram

U × {0} × L �� //

����

U × [0,∞)× L

��U // T,

where U is an open connected subset of some Euclidean space Rm and L isa height h − 1 sharp stratified space. We call U the local stratum, L the

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link and BT = U × [0,∞)× L the blowup. There are natural smooth mapsπ : T → U and ρ : T → [0,∞), and we call them the tubular projectionand the tubular function respectively. We use the adjectives latitudinal,radial and longitudinal when referring to the directions of U , [0,∞) and Lrespectively. Whenever 0 ∈ U we define the base point in T as the image of0 under the inclusion U → T at ρ = 0.

Having specified the allowed neighborhoods, we define sharp stratifiedspaces of height h as any Frolicher space that locally is diffeomorphic to atubular domain of height at most h. In addition we demand some topologicalstandard requirements.

Definition 3.2 A height h sharp stratified space W is a Frolicher spacesuch that

every point p in W has an open neighborhood V diffeomorphic to atubular domain T of height less than or equal to h,

some point p in W has an open neighborhood V diffeomorphic to atubular domain T of height exactly h, and

W is Hausdorff, paracompact, and has a countable basis for its topol-ogy.

Here are a few examples of sharp stratified spaces:

Example 3.3 i) Let M be any compact manifold. The cone of M is asharp stratified space.

ii) Let L be a discrete set with three points, {1, 2, 3}. Let R be the localstratum of a tubular domain T . Take any permutation σ of L. DefineW 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 aneighborhood diffeomorphic to the cone over four points.

iv) The Whitney umbrella can be constructed as a sharp stratified space bytaking the cone with link L the union of a figure eight and a disjointpoint.

Proposition 3.4 Sharp stratified spaces are balanced.

Proof: We must show that the functional and curved topologies areidentical. By induction assume that all links L are balanced. It followsfrom [Cap93, Proposition 1.16] that any blowup BT = U × [0,∞) × L also

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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 compositionBT → T ⊆ W . Then q−1(V ) is curved open, and by the induction hy-pothesis also functional open. For u ∈ U , ρ ∈ [0,∞) and z ∈ L define asmooth map g : BT → R by g(u, ρ, z) = |u|2 + ρ. Choose ε > 0 such thatK = g−1([0, ε]) becomes a compact subset of BT . Since K r q−1(V ) is com-pact 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 ≤ ε.Consequently, g−1(−δ, δ) is contained in q−1(V ). Observe that g descendsto a well-defined function f : T → R. Now x0 ∈ f−1(−δ, δ) ⊆ V ∩ T , andthis shows that V is a functional neighborhood of x0 in W . �

Since tubular domains are defined by pushout it is easy to study thesmooth functions of T . The main tool will be this version of Hadamard’slemma.

Lemma 3.5 (Hadamard) Given a smooth function f : T → R there areunique functions f0 : U → R and f1 : BT = U × [0,∞)× L→ R such that

f = f0π + ρf1.

Moreover, f0 and f1 depend smoothly on f .

By abuse of notation we will often omit π from Hadamards lemma andwrite f0 instead of f0π. With this simplification f = f0 + ρf1.

Proof: Let f0 be the composition of the canonical inclusion U → T withf . Observe that g = f − f0π is a smooth function on T that vanish on thelocal stratum U . Lift g to a smooth function g defined on the blowup BT .Since g(u, 0, z) = 0 for all z ∈ L the fundamental theorem of calculus gives

g(u, ρ, z) =

∫ 1

0

d

dtg(u, tρ, z)dt = ρ

∫ 1

0

∂g

∂ρ(u, tρ, z)dt.

Let f1(u, ρ, z) be∫ 1

0∂g∂ρ(u, tρ, z)dt. By construction f0 and f1 depend smoothly

on f . �

For manifolds there are several ways of defining the tangent space. Alter-natives include using curves, derivations or local one-parameter families ofdiffeomorphisms. These definitions generalize to our setting, but no longerdo they produce equivalent notions of tangent space. In order to tell whichstratum a point belongs to we will elaborate the curve approach to tangentspaces. They are called the kinematic tangent spaces in [Sta13, Section 4].

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Definition 3.6 The curve space CurvexX of a Frolicher space X at thepoint x is the equivalence classes of smooth curves c : X → R with c(0) =x under the relation that c1 ∼ c2 if (fc1)′(0) = (fc2)′(0) for all smoothfunctions f .

Proposition 3.7 Let W be a sharp stratified space. For all x ∈ W thereexists an integer m such that the curve space CurvexW is homeomorphicto Rm. Moreover, such x lies in the local stratum of a tubular domain withdimU = m.

Proof: Let us first prove that any point x ∈ W lies in the local stratumof some T . By induction on height we assume this to hold for any sharpstratified space L of lower height. Any point x ∈ W lies in some tubulardomain 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 tubulardomain T1 in L such that z lies in the local stratum U1. Then T containsa tubular domain U × (0,∞) × T1 where x lies in the new local stratumU × (0,∞)× U1.

Now assume that x lies in the local stratum U of some tubular domainT . We will identify the curve space CurvexW with the tangent space of Uat x. We compare a general curve c to its projection into the local stratumπc. Write f = f0 + ρf1 by Hadamard. We have f0c = fπc.

Let C be a compact neighborhood of x in U and let δ > 0. Since L iscompact there is an upper bound K on f1 restricted to C × [0, δ] × L. Forsufficiently small t the curve c(t) lies under C × [0, δ]×L and it follows that

|fc(t)− fπc(t)| ≤ K · ρc(t).

Therefore

0 ≤ |(fc)′(0)−(fπc)′(0)| = limt→0

∣∣∣∣fc(t)− fπc(t)t

∣∣∣∣ ≤ limt→0

K·ρc(t)t

= K·(ρc)′(0).

Since ρ ≥ 0 we have a minimum for ρc(t) at t = 0. Hence (ρc)′(0) = 0 andconsequently (fc)′(0) = (fπc)′(0) for all smooth f . It follows that c and πcare equivalent in the curve space. Thus CurvexW ∼= TxU . �

There are a few axioms related to stratifications of topological spaces.See [Mat70]. A stratification of X is a cover by pairwise disjoint smoothmanifolds {Si}. We say that the stratification is locally finite if each pointof X has a neighborhood that meets at most finitely many strata. Theaxiom of the frontier is satisfied if the topological boundary of a stratum Siin 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 CurvexW .

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Definition 3.8 A stratum S of dimension m in a sharp stratified space Wis a connected component of the inverse image dim−1(m). The dimensionof W is defined as the maximal m such that W has a non-empty stratum ofdimension m.

Proposition 3.9 The strata {S} of W is a stratification. It is locally finiteand 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 tubulardomain T , where x lies in the local stratum, and then U ⊆ S is a locallyEuclidean neighborhood of x.

By induction on height we assume that the stratification of sharp strati-fied spaces of height lower than W is locally finite. To test locally finitenessfor W we will argue that any T meets finitely many strata. The strata of Tare U together with U × (0,∞) × S for each stratum S of the link L. ButL is compact, so the induction hypothesis implies that L has only finitelymany strata. Thus T meets only finitely many strata of W .

Let S be a fixed stratum and consider another stratum S′ that meets theboundary of S. For a contradiction assume that S′ also contains points out-side the boundary of S. Since S′ is connected there is some tubular domainT such that U ⊆ S′ and U contains points both inside ∂S and outside ∂S.But this is impossible since the intersection T∩S has the form U×(0,∞)×Ywhere Y is a union of strata in the link L. Consequently S′ is completelycontained 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 Fro(W,Rn).We will use a theorem by Frolicher and Kriegl [FK88, Theorem 6.1.4],also [KM97, Theorem 4.11]. It says that given a Frechet space structuresuch that the Frechet smooth curves are the Frolicher smooth curves, thenthe Frechet topology equals the c∞-topology, i.e. the curved topology.

Recall that a Frechet space E is a vector space with a countable familyof seminorms, ‖−‖n, such that E is Hausdorff and complete, see [Ham82]. IfU ⊆ Rk is open, then the space of smooth functions C∞(U,Rn) is a Frechetspace with seminorms

‖φ‖α,K = supx∈K‖∂αφ(x)‖,

where K runs through a countable family of compact subsets covering U ,and ∂α denotes the higher order derivative corresponding to the multi-index

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α. By Cartesian closedness, see [KM97, Theorem 3.12], a curve c : R →C∞(U,Rn) is smooth as a map into a Frechet space if and only if the adjointc : R×U → Rn is a smooth map. Consequently, the Frechet space structureon C∞(U,Rn) agree with the Frolicher space structure.

Let us inductively define seminorms on Fro(W,Rn). By induction onheight ofW , we assume that there is a Frechet space structure on Fro(W ′,Rn)for all W ′ of height less than W .

Definition 4.1 Let T ⊆W be a tubular domain. Let ‖− ‖i be a seminormon C∞(U,Rn) and let ‖−‖j be a seminorm on Fro(BT ,Rn). Then we defineseminorms ‖−‖πT,i and ‖−‖BT,j on Fro(W,Rn) by using the Hadamard lemmamapping φ ∈ Fro(W,Rn) by the composition

Fro(W,Rn)→ Fro(T,Rn) ∼= C∞(U,Rn)×Fro(BT ,Rn)

to a pair (φ0, φ1). With this notation we define

‖φ‖πT,i = ‖φ0‖i and ‖φ‖BT,j = ‖φ1‖j .

The key ingredient in establishing a Frechet structure on Fro(W,Rn) iscomparing structures on two intersecting tubular domains. To do this weneed a technical result:

Proposition 4.2 Let T and T ′ be tubular domains in W both based at x0.There exists a compact neighborhood C of 0 in U , δ > 0 and a smoothmap η : C × [0, δ] × L → U ′ × [0,∞) × L′ commuting with projection intoW . Moreover, the ratio between the tubular functions r

ρ is a strictly positivesmooth map on C × [0, δ]× L.

Proof: We choose the compact neighborhood C and δ > 0 such that theimage of C × [0, δ]× L in W lies inside T ′. Away from the local strata theidentity of W defines η smoothly from C × (0, δ] × L to U ′ × (0,∞) × L′.In order to extend η to a smooth map defined on C × [0, δ]× L we have toshow that the three projections

ηlat : C × [0, δ]× L→ U ′

ηrad : C × [0, δ]× L→ [0,∞)

ηlong : C × [0, δ]× L→ L′

exist and are smooth. The first two projections, ηlat and ηrad exist and aresmooth because they are restrictions of the tubular projection π′ : T ′ → U ′

and the tubular function r : T ′ → [0,∞) respectively. It remains to showthat ηlong extends smoothly.

Since the tubular function r : T ′ → [0,∞) is smooth, we may applyHadamard’s lemma in T and write r = ρ · r, where r is a smooth function

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defined on C × [0, δ]× L. We claim that r > 0 for all points in its domain.If not, there is a u0 ∈ U and z0 ∈ L such that limn→∞ r(u0,

1n , z0) = 0.

Since u0 also lies in the local stratum U ′, we may use Hadamard’s lemmain T ′ near u0 to write ρ = rρ, where ρ is smooth on a neighborhood of{u0}× 0×L′ in U ′ × [0,∞)×L′. Back in C × (0, δ]×L we get the identity

ρ = rρ = ρrρ

at least for points (u, t, z), with t > 0, sufficiently near (u0, 0, z0). Dividingboth sides by ρ we get

r(u, t, z) · ρ(η(u, t, z)) = 1

The link L′ is compact, thus the sequence ηlong(u0,1n , z0) has an accumu-

lation point z′ in L′. Since ρ(u0, 0, z′) is finite, it is an impossibility that

limn→∞ r(u0,1n , z0) = 0. This proves the claim that r > 0 everywhere.

A priori ηlong is at least defined on C × (0, δ] × L. Let f ′ : L′ → R bea smooth function. We will now show that the composition f ′ηlong extendsto a smooth function on C × [0, δ] × L′. This implies that ηlong extendssmoothly to C × [0, δ]× L′ and will thus complete the proof.

Clearly the assignment (u, r, z′) 7→ r · f ′(z′) defines a smooth functionon T ′. Applying Hadamard’s lemma in T yields

r · f ′ = ρ · f

where f is a smooth function defined on C × [0, δ] × L. Solving for f ′ηlong

and using r = ρr, we get

f ′ηlong =f

r.

Moreover, fr is smooth on C× [0, δ]×L, since f and r are smooth and r > 0.Thus f ′ηlong extends uniquely to a smooth function on C × [0, δ] × L, andwe are done. �

For Siebenmann’s locally conelike TOP stratified sets there are examplesderived from Milnor’s counterexamples to the Hauptvermutung [Mil61] thatshow that the link is not unique up to homeomorphism. See the remarkin [Sie72] after his Definition 1.2. In the category of Frolicher spaces wemay 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 ′ based at x0 we get smooth maps

η : C × [0, δ]× L→ U ′ × [0,∞)× L′ and

η′ : C ′ × [0, δ′]× L′ → U × [0,∞)× L

12

above the identity of W . It follows that η′ is the inverse of η away from thelocal stratum, but then η−1 = η′ also in the limit ρ → 0. Restricting overthe basepoint x0 gives a diffeomorphism L ∼= L′. �

To construct the Frechet structure we proceed as follows: Given W , letT be the category of tubular domains, where the local stratum U , link L andmap BT → T is part of the structure. The morphisms in T are inclusionsT ′ ⊆ T as subsets of W . By induction on height we have for each T a Frechetspace structure on Fro(T,Rn) ∼= C∞(U,Rn)× Fro(BT ,Rn). We propose toconstruct the Frechet structure on Fro(W,Rn) as the limit of Fro(T,Rn)over the category T . Clearly the limit is complete when each Fro(T,Rn) iscomplete. The following lemmas check that this construction is well-defined:

Lemma 4.4 An inclusion T ⊆ T ′ induce a continuous linear map of Frechetspaces Fro(T ′,Rn)→ Fro(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 ofboth T and T ′. Thus U maps into U ′ by a local diffeomorphism h : U → U ′.

Using the Hadamard lemma in T ′ and T we get a map

C∞(U ′,Rn)×Fro(BT ′ ,Rn)→ C∞(U,Rn)×Fro(BT ,Rn)

by sending (φ′0, φ′1) to (φ0, φ1), such that the equation φ0 + ρφ1 = φ′0 + rφ′1

holds in T . Observe that the map splits as a product. We have φ0 = h∗(φ′0)where h∗ : C∞(U ′,Rn)→ C∞(U,Rn) is continuous by the chain rule.

To check continuity of φ′1 7→ φ1 we apply proposition 4.2 to get a smoothmap

η : C × [0, δ]× L→ U ′ × [0,∞)× L′

covering the inclusion T ⊆ T ′ in a neighborhood of x0. Then φ1 is the imageof φ′1 under the composition of three continuous maps: The multiplicationby the tubular function r of T ′ is a map

Fro(BT ′ ,Rn)r·−→ Fror=0(BT ′ ,Rn),

where the target consists of smooth functions vanishing on the local stratumU ′. By induction on height of W we already know that the map

Fror=0(U ′ × [0,∞)× L′,Rn)η∗−→ Froρ=0(C × [0, δ]× L,Rn)

induced by η is smooth, hence continuous. The last map is the inverse ofmultiplication by r, and it is smooth, hence continuous, by Hadamard’slemma:

Froρ=0(C × [0, δ]× L,Rn)Hadamard−−−−−−→ Fro(C × [0, δ]× L,Rn)

13

This proves the lemma. �

Lemma 4.5 The limit of Fro(T,Rn) is metrizable, i.e. there is a countablefamily of seminorms defining the topology.

Proof: Consider the functor Fro(−,Rn) as a sheaf of locally convex com-plete vector spaces on W . Since W has a countable covering by tubulardomains {T i}, we get that Fro(W,Rn) is a closed subspace of the countableproduct

∏iFro(Ti,Rn). Consequently Fro(W,Rn) is metrizable. �

Proposition 4.6 There is a countable family of the seminorms on Fro(W,Rn)of the type described definition 4.1. Thus we have a Frechet space. More-over, the smooth curves c : R→ Fro(W,Rn) defined by the Frechet structureare precisely the Frolicher smooth curves. Consequently, the curved topologyon Fro(W,R) equals the Frechet topology.

Proof: The only part remaining of the proof is to compare the Frechet andFrolicher structures. Let c : R → Fro(W,Rn) be Frechet smooth. Considera tubular domain T and decompose c = c0 + ρc1. By Cartesian closednessfor the local stratum U , we get that c0 is Frechet smooth if and only if theadjoint c0 : R×U → Rn is smooth, i.e. c0 is Frolicher smooth. By inductionwe also have that c1 is Frechet smooth if and only if it is Frolicher smooth.Running through all T in a covering of W , we get that that Frechet andFrolicher smoothness coincide.

The last statement follows from Frolicher 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 neigh-borhoods in the curved topology on Fro(W,Rn).

5 Coordinate systems

In order to study embeddings of sharp stratified spaces it is crucial to un-derstand the smooth structure in a neighborhood of each stratum. Basicallywe want to use calculus, so we need coordinate systems. Features in the linkshrink 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 tubulardomains (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 someEuclidean space and has coordinates which we will denoted by

uj = (uj1, . . . , ujm).

The blowup of the coordinate system is the Frolicher space

Bu = U0 × [0,∞)× · · · × Uk × [0,∞)× Lk.

There is a natural map Bu → W . In the latitudinal direction we have thelower strata U0× [0,∞)× · · · ×Uk. The k-th tubular function ρk representsthe 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 uis for each j a compact neighborhood Cj of 0 in Uj and a δj > 0. Thus weget 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 ΠU forU0 × (0,∞)× · · · × Uk. Similarly, given a core, we use the notation ΠC forC0 × [0, δ0]× · · · × Ck. If we need to emphasize the depth we write ΠkU orΠkC.

On a coordinate system u of depth k we may apply Hadamard’s lemmarepeatedly to any smooth function f . First step gives f = f0 + ρ0f1 wheref0 : U0 → R. Applying the lemma to f1 we get f1 = f1 + ρ1f2 wheref1 : U0 × [0,∞)× U1 → R is smooth. Inductively we get

f = f0 + ρ0f1 + ρ0ρ1f2 + . . .+ ρ0 · · · ρk−1fk + ρ0 · · · ρkfk+1,

where fj are ordinary smooth functions of several variables with domain

U0 × [0,∞)× · · · × Uj = ΠjU and fk+1 is smooth on Bu.

Definition 5.2 The Hadamard expansion of f in a coordinate system u ofdepth k is

Hkf = f0 + ρ0f1 + ρ0ρ1f2 + . . .+ ρ0 · · · ρk−1fk.

We call ρ0 · · · ρkfk+1 the remainder.

For all coordinates, uji and ρj , there are partial derivatives ∂f∂uji

and ∂f∂ρj

.

Differentiating a Hadamard expansion, say f = f0 + ρ0f1 + ρ0ρ1f2, withrespect to a coordinate of depth > 0, say u11, we get

∂f

∂u11= ρ0

∂f1

∂u11+ ρ0ρ1

∂f2

∂u11.

15

Observe that ρ0 can be factorized on the left hand size. This means that ∂f∂u11

vanish on the stratum where ρ0 = 0. To get information from this partialderivative at the stratum ρ0 = 0, we will rescale and consider 1

ρ0∂f∂u11

, andsimilar expressions, instead. Therefore we introduce the scaled gradient andscaled Jacobi matrix.

Definition 5.3 Let f be a smooth function of W . With respect to a depthk coordinate system u on W we define the scaled gradient vector to be thevector Su

kf with entries

1

ρ0 · · · ρj−1

∂f

∂ujiwith j ≤ k and

1

ρ0 · · · ρj−1

∂f

∂ρjwith j < k.

Similarly, for smooth maps φ : W → Rn we have the scaled Jacobi matricesSukφ.

Proposition 5.4 The scaled Jacobi matrix Suk is a continuous map defined

on Fro(W,Rn)×Bu.

Proof: Consider the depth j − 1 Hadamard expansion with remainder

f = f0 + ρ0f1 + . . .+ ρ0 · · · ρj−2fj−1 + ρ0 · · · ρj−1fj .

Only the remainder is non-constant with respect to uji and ρj . Differenti-ating we get

∂f

∂uji= ρ0 · · · ρj−1

∂fj∂uji

, and

∂f

∂ρj= ρ0 · · · ρj−1

∂fj∂ρj

.

Since∂fj∂uji

and∂fj∂ρj

are smooth onBu and depend smoothly on f in Fro(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. Amixed product has the form c0 · · · ck where ci is either ρi or ri. The weightedmean of mixed products pk is defined as the integral

pk =

∫ 1

0(tρ0 + (1− t)r0) · · · (tρk + (1− t)rk) dt.

16

Calculations of the first few weighted products are

p0 = 12ρ0 + 1

2r0

p1 = 13ρ0ρ1 + 1

6ρ0r1 + 16r0ρ1 + 1

3r0r1

p2 = 14ρ0ρ1ρ2 + 1

12ρ0ρ1r2 + 112ρ0r1ρ2 + 1

12r0ρ1ρ2

+ 112ρ0r1r2 + 1

12r0ρ1r2 + 112r0r1ρ2 + 1

4r0r1r2.

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 kcoordinate 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, . . . , ρjand r0, . . . , rj.

Theorem 5.7 (A scaled mean value theorem) Let f : W → R be asmooth function and let u be a depth k coordinate system. Precomposi-tion with the canonical map ΠU → W gives the map Hkf , defined on thelower strata. Let x and y be points in the lower strata ΠU of u such that allweighted means of mixed products are non-zero, i.e. pk−1 6= 0. It is possibleto choose a family of vectors Mf defined for (x, y) ∈ (ΠU×ΠU)r{pk−1 = 0}and depending smoothly on f such that

Hkf(y)−Hkf(x) = Mf · Λk(x, y)

and the limit of Mf whenever x and y converge to the same point x0 ∈ ΠUis Su

kf(x0). More precisely, for all ε > 0 there is a neighborhood U of f inFro(W,Rn) and a compact neighborhood K of x0 in ΠU and g ∈ U such that

|Mg − Sukf(x0)| < ε,

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 havingthe form of a Hadamard expansion, see Definition 5.2. A standard proof forthe mean value theorem starts by rewriting the difference f(y) − f(x) as aline integral

f(y)− f(x) =

∫C∇f · dc,

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 rewritethe integral as

=

∫ 1

0M(t) · Λk(x, y) dt,

where M(t) is the vector field along C with coordinates

1

pj−1

∂f

∂uji

∣∣c(t)

and1

pj−1

∂f

∂ρj

∣∣c(t).

Clearly Mf =∫ 1

0 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 showthat the scaled gradient vectors continuously extend the definition of Mf topoints where x = y. More precisely, the assignment

(x, y) 7→

{Mf for pk−1 6= 0

Sukf(y) for x = y

is continuous.In the lemma below we will show that∫ 1

0

1

pj−1

∂f

∂uji

∣∣c(t)

dt→ ∂fj∂uji

(x0) and∫ 1

0

1

pj−1

∂f

∂ρj

∣∣c(t)

dt→ ∂fj∂ρj

(x0)

when x and y tends to x0. Here fj is the remainder in the Hadamard ex-pansion of depth j. The limits are the coordinates of Su

kf(x0), so the proofis complete modulo the lemma. �

Lemma 5.8 Let x and y be elements in a compact and convex neighborhoodof x0 in ΠU . There exists a neighborhood U of f in Fro(W,R) and a constantC such that ∣∣∣∣∫ 1

0

1

pj−1

∂g

∂uji

∣∣c(t)

dt− ∂gj∂uji

(y)

∣∣∣∣ ≤ C|x− y|18

for g ∈ U . Here gj is the remainder in the Hadamard expansion. A similarstatement holds for the partial derivative in ρj-coordinates.

Proof: Differentiating the Hadamard expansion we get

∂g

∂uji= ρ0 · · · ρj−1

∂gj∂uji

.

Thus we may rewrite the integral as∫ 1

0

1

pj−1(tρ0 + (1− t)r0) · · · (tρj−1 + (1− t)rj−1)

∂gj∂uji

∣∣c(t)

dt.

Using integration by parts with one factor being P (t) with P ′(t) = (tρ0 +(1− t)r0) · · · (tρj−1 + (1− t)rj−1) and P (0) = 0, we get

=

[P (t)

pj−1

∂gj∂uji

]1

0

−∫ 1

0

P (t)

pj−1∇(∂gj∂uji

)· (x− y) dt.

Since P (t) is increasing and defines the weighted mean of mixed products by

P (1) = pj−1, we observe that P (t)pj−1

lies in the range [0, 1]. Moreover, we can

choose the neighborhood U such that ∇(∂gj∂uji

)is bounded on the convex

subset of the lower strata. By the mean value theorem for integrals there isa vector ~v such that we get

=P (1)

pj−1

∂gj∂uji

(y)− ~v · (x− y).

where the entries of ~v have the form K∂2gj∂ξ∂uji

(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 struc-ture between strata of stratified sets. In this section we will consider smoothmaps φ : W → Rn that topologically are embeddings. When all scaled Ja-cobi matrices Su

kφ have linearly independent column vectors, we can showthat the image φ(W ) satisfies the Whitney conditions.

Definition 6.1 A smooth map φ : W → Rn is called an S-immersion ifall depth k coordinate systems u have scaled Jacobi matrices Su

kφ(x) whosecolumn vectors are linearly independent at any point x ∈ Bu.

19

Whenever φ : W → Rn is an S-immersion and topologically an embed-ding, then the image φ(W ) inherits a stratification from W . If {S} is the setof strata in W , then {φ(S)} is locally finite and satisfied the axiom of thefrontier because W and φ(W ) are homeomorphic. When an S-immersion φis restricted to a stratum S, we get an ordinary immersion of smooth man-

ifolds Sφ−→ 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 Rnsuch that X lies in the boundary of Y . Let {yi} be a sequence in Y convergingto some point x ∈ X. Suppose that the tangent spaces TyiY converge to someτ . We say that the pair of strata X and Y satisfies Whitney condition A ifthe situation above always implies that TxX ⊆ τ .

Definition 6.3 (Whitney condition B) Let X and Y be strata such thatX lies in the boundary of Y . Consider sequences {yi} in Y and {xi} in Xsuch that both sequences converge to x0 ∈ X. Assume that the tangent planesTyiY converge to some hyperplane τ , and that the secant lines from xi to yiconverge 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 anS-immersion. Consider strata X and Y of φ(W ) such that X is containedin the closure of Y . Given a sequence {yi} in Y converging to x0 ∈ X thereis 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 inU0 × (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 suf-ficiently large i each yi lies in T and corresponds uniquely to a point inU0× (0,∞)×L. The projection into L forms a sequence {yi}, and since thelink is compact we may choose a subsequence {yi}i∈I1 converging to somepoint z0 in L. By induction hypothesis there is a depth k − 1 coordinatesystem u in L based at z0 together with a subsequence {yi}i∈I2 convergingto 0 in U1 × [0,∞) × · · · × Uk. Now use T and u to construct a coordi-nate system u in W based at x0. The subsequence {yi}i∈I2 converge to 0 inU0 × [0,∞)× · · · × Uk. �

20

Let us use the scaled Jacobi matrices to recognize tangent spaces ofstrata. We will use the following notation. Given a depth k coordinate

system u our map φ gives a map Buφ−→ Rn. Over the blowup

Bu = U0 × [0,∞)× · · · × Uk × [0,∞)× Lk

we define the depth k tangent space as

T kBu = Bu × Rm

where m is the dimension of U0 × [0,∞)× · · · × Uk. By construction of thescaled Jacobi matrix we get a commutative diagram

T kBu

Sukφ //

����

TRn

����Bu

φ // Rn.

Given a point x ∈ Bu let T kxBu denote the depth k tangent vectors based atx. Let φ∗T

kxBu be the image under the scaled Jacobi matrix. Geometrically

consider the stratum Y ⊆ Rn parameterized under φ by the points x withρk = 0 and other ρj 6= 0. The vector spaces φ∗T

kxBu are the tangent spaces

of Y for ρk = 0, and they extend the tangent spaces to a continuous familyfor 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 alsois an S-immersion. Let Y be a stratum of φ(W ) and consider a depth kcoordinate 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 thelower strata ΠU , then TyiY converge to the depth k tangent space φ∗T

k0 Bu.

Proof: Rescaling the Jacobi matrix does not change the column space, soTyiY

∼= φ∗TkyiBu. We have assumed that Su

kφ(0) has linearly independentcolumn vectors. Therefore the rank of Su

kφ does not fall when yi converge to0 in ΠU . By continuity of Su

kφ it follows that φ∗TkyiBu converge to φ∗T

k0 Bu. �

This immediately implies that φ(W ) satisfies Whitney condition A.

Corollary 6.6 Let φ : W → Rn be a topological embedding and an S-immersion. If X and Y are strata of φ(W ) such that X is contained inthe 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. ThenTyiY converge to τ = φ∗T

kx0Bu. Since the tangent space of X at x0 is

φ∗T0x0Bu0 , it follows that Tx0X ⊆ τ . Hence Whitney condition A holds. �

Proposition 6.7 Let φ : W → Rn be a topological embedding and an S-immersion. If X and Y are strata of φ(W ) such that X lies in the boundaryof Y , then they satisfies Whitney condition B.

Proof: By the lemma above choose a depth k coordinate system u suchthat 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 Hkφ(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)

|φ(yi)− φ(xi)|= Mφ · ~vi.

Here the left side is a unit vector converging to a direction vector for theline `.

Since all Sukφ have linearly independent column vectors there exists a

constant a > 0 and a neighborhood of matrices M around Sukφ(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 inthis neighborhood. Now |Mφ~vi| = 1 implies that |~vi| ≤ 1

a . By compactnessit is possible to choose a subsequence such that {~vi}i∈I2 converge to somevector ~v. By continuity, the limit of Mφ~vi is Su

kφ(0)~v. Moreover ~v 6= 0 since1 = |Mφ~vi| = |Su

kφ(0)~v|. Proposition 6.5 says that Sukφ(0) spans τ . We have

shown that Sukφ(0)~v is a direction vector for `. It follows that ` ⊆ τ . �

Remark 6.8 Cherenack conjectures, see [BC05, Conjecture 4.1], that theupper half cone x2 +y2 = z2, z > 0 is Frolicher diffeomorphic to the quotientof S1 × [0,∞) by collapsing S1 × {0} to a point. However, Bentley andCherenack were unable to prove this. We can generalize this question and

22

ask if φ : W → Rn gives a Frolicher diffeomorphism from W onto its imagewhenever φ is a topological embedding and an S-immersion. Here φ(W ) isgiven the smooth structure of a subset in Rn.

However, the generalized conjecture is certainly false. The cone C on theinterval [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 tangentplane. Whereas any distribution λ : C∞([0, π2 ],R) → R gives rise to a dis-tinct derivation at the base point of the cone C.

Nevertheless, if W is a sharp stratified space and W ′ is a coarser smoothstructure on W , then the image of φ′ : W ′ → Rn is still a Whitney stratifiedset whenever the pullback φ to W ia a topological embedding and an S-immersion.

7 Change of coordinates and the open set of S-immersions

In this section we will study how a change of coordinates affects the scaledJacobi matrices. It turns out that linear independence of column vectors inSkφ does not depend on the choice of coordinate system. We use this factto show that S-immersions of a compact sharp stratified space W form anopen set in Fro(W,Rn).

Translation lets us compare coordinate systems with different basepoints,but it remains to discuss coordinate systems with the same basepoint. Wewill 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 coordinatesystems 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 coordinatesystemes v have scaled Jacobi matrices Sv

l φ with linearly independent columnvectors, then φ is an S-immersion.

Proof: Any u not of maximal depth may be extended to some v of maximaldepth. Since all columns of Su

kφ also are columns in Svl φ the result follows. �

Definition 7.2 Let u and v be max depth coordinate systems in W . Wesay that u and v have the same basepoint if

• the tubular domains T0 and T ′0 have the same basepoint in W , and

• for all j we identify the links Lj−1 and L′j−1 such that Tj and T ′j havethe same basepoint in Lj−1

∼= L′j−1.

23

Let us now consider the scaled Jacobi matrix in two max depth coordi-nate systems with the same basepoint.

Proposition 7.3 Let φ : W → Rn be a smooth map. Let u and v be maxdepth coordinate systems with the same basepoint. Then the column spacesare equal:

ColSukφ(0) = ColSv

kφ(0)

Proof: By induction on height we may assume that the statement holdsfor all links in W . Given max depth coordinate systems u and v we con-sider the change of coordinate map of the bottom tubular domains given byproposition 4.2:

η : C0 × [0, δ]× L→ U ′ × [0,∞)× L′.

Use Hadamard’s lemma to write φ in both T0 and T ′0. We get

φ = φ0 + ρ0φ1 = φ0 + r0φ′1.

From Proposition 4.2 we know that the quotient r0 = r0ρ0

is smooth and

non-negative as a function on C × [0, δ]× L. Thus φ1 = r0 · φ′1.Write u for the coordinate system in L consisting of the last tubular do-

mains (T1, . . . , Tk) from u. By Hadamard’s lemma we may block decomposethe scaled Jacobi matrix over ρ0 = 0 as

Sukφ =

(∂φ0

∂u0i

∣∣∣∣ φ1

∣∣∣∣ S uk−1φ1

).

Compare this to the proof of Proposition 5.4. Differentiating φ1 = r0 · φ′1 bythe product rule, we get

S uk−1φ1 =

(S uk−1r0

)φ′1 + r0 · S u

k−1φ′1.

By induction the column space of S vk−1φ

′1 equals that of S u

k−1φ′1. Moreover,

by the chain rule the column spaces of(∂φ0∂u0i

)and

(∂φ0∂v0i

)are equal. Thus

we see that

Svkφ =

(∂φ0

∂v0i

∣∣∣∣ φ′1 ∣∣∣∣ S vk−1φ

′1

)has the same column space as Su

kφ. �

Let us first consider the passage from a coordinate system u into thestratum where some ρs 6= 0.

24

Definition 7.4 Let u be a depth k based coordinate system. An elementarychange of stratum is the following construction of a new based coordinatesystem v where the old tubular function ρs, for some s, becomes an ordinarycoordinate. The new tubular domains are given by

T ′j = Tj for j < s,

T ′s∼= Us × (0,∞)× Ts+1 and

T ′j = Tj+1 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 elementarychange of stratum of a based depth k coordinate system u. Then there is afiberwise invertible map P such that the following diagram commutes

T k−1BvP //

����

Svk−1φ

$$T kBu

Sukφ //

����

TRn

����Bv

⊆ // Buφ // Rn.

Consequently φ∗Tk−1x Bv = φ∗T

kxBu for all x ∈ Bv.

Proof: The difference between the coordinate systems u and v is that thetubular 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 theentries of Su

kφ are

1

ρ0 · · · ρj−1

∂f

∂ujiwith j ≤ k and

1

ρ0 · · · ρj−1

∂f

∂ρjwith j < k.

Similarly Svk−1φ has entries

1

r0 · · · rj−1

∂f

∂vjiwith j ≤ k − 1 and

1

r0 · · · rj−1

∂f

∂rjwith j < k − 1.

25

We have for j > s

1

r0 · · · rj−1

∂f

∂vji= ρs

1

ρ0 · · · ρj−1

∂f

∂ujiand

1

r0 · · · rj−1

∂f

∂rj= ρs

1

ρ0 · · · ρj−1

∂f

∂ρj.

Otherwise, i.e. for j ≤ s, the entries in the scaled Jacobi matrices are equal.Consequently

Svk−1φ = Su

kφ · P,

where P is a diagonal matrix with 1 on the diagonal for columns correspond-ing to j ≤ s and ρj on the entries corresponding to j > s. In the domain ofv all ρj 6= 0, thus P is invertible. The result follows. �

For fixed a coordinate system v we will be interested in nearby coordinatesystems of various depths.

Definition 7.6 Consider a coordinate system v of depth l together with acore consisting of compact neighborhoods C ′j ⊆ Vj and constants δ′j > 0. Wesay that a max depth coordinate system u is based in the core of v if thereis a smooth map

η : C0 × [0, δ0]× · · · × [0, δk]× Ck → V0 × [0,∞)× · · · × Vl × [0,∞)× L′l

above the identity of W , where each Cj is a compact neighborhood of 0 inUj, and the map η sends the basepoint to η(0) inside the core given by C ′jand δ′j for j = 0, . . . , l.

Proposition 7.7 Let φ : W → Rn be an S-immersion. Given a depth lcoordinate system v there exists a neighborhood U of φ in Fro(W,Rn) and acore such that for any ψ ∈ U and any max depth coordinate system u basedin the core of v the scaled Jacobi matrix Su

kψ(0) has linearly independentcolumn vectors.

Proof: By downward induction of the depth of v. For a maximal depthcoordinate system v we know by continuity of the scaled Jacobi matrixdefined over Fro(W,Rn)×Bv that there exists a neighborhood U of φ and acore C ′0× [0, δ′0]×· · ·×C ′l such that Sv

l ψ(y) has linearly independent columnvectors for all ψ ∈ U and y in the core.

For the start of the induction assume that u is based in the core of themax depth coordinate system v. Then there exists a chain of elementarychanges of strata w0 = v, w1, . . ., wp such that wp has the same basepointas u. By our previous results we have

ψ∗Tk0 Bu = ψ∗T

l−pη(0)Bwp = · · · = ψ∗T

lη(0)Bw0 = ψ∗T

lη(0)Bv.

26

Consequently Sukψ(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

L′ of v choose a tubular domain T ′z ⊆ L′ based at z. Let V zl be the local

stratum of T ′z and denote the link by Lz. For each z we get a depth lcoordinate system vz given by the natural map

V0 × [0,∞)× · · · [0,∞)× Vl−1 × [0,∞)× V zl × [0,∞)× Lz →W.

By the induction hypothesis there is a neighborhood Uz of φ in Fro(W,Rn)and a core of vz specified by C ′z0 , . . . , C

′zl and δ′z0 , . . . , δ

′zl > 0 such that

Sukψ(0) has linearly independent column vectors for all coordinate systems

u based in the core of vz. Let Pz be the image of C ′zl × [0, δ′zl ] × Lz in the

link L′. The interiors Pz give an open covering of the compact space L′, sowe may choose a finite subcovering indexed by z ∈ I.

Intersection U =⋂z∈I Uz gives a neighborhood of φ in Fro(W,Rn) to-

gether with a core of v specified by C ′j =⋂z∈I C

′zj and δ′j = minz∈I δ

′zj where

j = 0, . . . , l − 1. Consider a max depth coordinate system u based in thiscore 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,∞)× L′.

The projection of η(0) to the link L′ lies in the open patch Pz 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,∞)× V zl × [0,∞)× Lz.

Consequently, u is based in the core of vz. Then by the induction hypoth-esis the scaled Jacobi matrix Su

kψ(0) will have linearly independent columnvectors 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 S-immersions in Fro(W,Rn) is open.

Proof: Let φ : W → Rn be an S-immersion. For all z in W choose atubular domain Tz based at z. Each Tz corresponds to a coordinate systemvz, so there is a neighborhood Uz of φ and a core C ′z0 , δ′z0 > 0 in vz suchthat for any u based in the core we have a scaled Jacobi matrix Su

kψ(0) withlinearly independent column vectors.

The patches below C ′z0 × [0, δ′z0 ] × Lz cover W and we choose a finitesubcovering 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. ThusSukψ(0) has linearly independent column vectors. �

8 The open set of S-embeddings

Although we know that injective S-immersions have a Whitney stratifiedimage in Rn, we have not yet described an open subset of Fro(W,Rn) ofmaps with this property. In our approach we will define S-embeddingsby insisting on injectivity not only for W , but also injectivity for all linkslatitudinally in the limit when ρ→ 0. In our approach we consider a depthk coordinate system. For the longitudinal direction we use ordinary calculusvia the Hadamard expansion. The longitudinal direction corresponds to theremainder 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 basedcoordinate system. Consider a point x in the lower strata ΠU . Denote byVx the orthogonal complement of φ∗T

kxBu. Define the radial map φ⊥x to be

the composition

Lkφk //

φ⊥x

66Rn pr // Vx.

Definition 8.2 Let φ : X → V be a smooth map from a Frolicher spaceinto 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 amanifold, then the S-embeddings φ : W → Rn are defined to be the ordinaryembeddings of manifolds. Suppose inductively that we have specified the S-embeddings Lk → Rm for all links occuring in W . For normed vector spacesV we have the usual notion of a normalization map N : V r 0 → V givenby N(~v) = ~v

|~v| . 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 S-embedding at x ∈W if φ is an S-immersion, and for every tubular domainT based at x the radial map φ⊥0 at the basepoint is ray injective and thecomposition with normalization is an S-embedding

Nφ⊥0 : L→ V0.

We say that φ is a longitudinal S-embedding if it is a longitudinal S-embedding 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 φis an injective longditudinal S-embedding.

Lemma 8.5 Let W be a compact sharp stratified space. If the set of S-embeddings in Fro(W,Rn−1) is open and dimW < n − 1, then the set ofray injective maps φ in Fro(W,Rn) such that Nφ is an S-embedding is alsoopen.

Proof: Let U0 be the set of maps φ in Fro(W,Rn) such that φ(x) 6= 0 for allx ∈W . Since W is compact it follows that U0 is open. Let V be the subsetof U0 consisting of ray injective φ such that Nφ are S-embeddings. Normal-ization induces a continuous map N∗ : U0 → Fro(W,Sn−1) where Sn−1 isthe sphere of unit vectors in Rn. For ~v ∈ Sn−1 let U~v be the subset of mapsW → Sn−1 whose image does not contain ~v. Each U~v is open, and togetherall U~v cover Fro(W,Sn−1) since dimW < n−1. By stereographic projectionthere are isomorphisms U~v ∼= Fro(W,Rn−1). Given a ray injective map φin Fro(W,Rn) such that Nφ is an S-embedding there is a ~v such that Nφlies in the open set V~v of S-embeddings in U~v ∼= Fro(W,Rn−1). The inverseimage of V~v under stereographic projection and N∗ is a neighborhood of φinside V. �

Lemma 8.6 Let φ : W → Rn be an S-immersion where dimW < n. As-sume that the sets of S-embeddings in Fro(L,Rm) are open for all links Loccurring in W and m > dimL+ 1. If φ is a londitudinal S-embedding at xand T is a tubular domain based at x, then there exists a core and a neigh-borhood U of φ in Fro(W,Rn) such that for all y in the patch P below thecore C × [0, δ]×L and ψ ∈ U we have that ψ is a longitudinal S-embeddingat y.

Proof: By downward induction on depth we will prove the following. Givena depth k coordinate system u based at x there is a neighborhood U of φand a core such that for all y in the corresponding patch ψ is a longitudinalS-embedding at y. The claim of the lemma is the statement for k = 0. Thestatement holds trivially for max depth coordinate systems since they havetrivial links.

For the induction step consider some u based at x and with depth k.Let Vx be the orthogonal complement of φ∗T

kxBu. Let G be the Grassma-

nian of dimVx subspaces V in Rn. Consider the associated bundle Ek → Gwith fibers Fro(Lk, V ). Let US-imm be the open set of S-immersions in

29

Fro(W,Rn). The construction of the radial map gives us a continuous map-ping

radialk : US-imm × U0 × [0,∞)× · · · × Uk → Ek.More precisely radialk(ψ, y) = ψ⊥y . Let V be the subset of Ek consistingof ξ : Lk → V that are ray injective and have normalizations Nξ that areS-embeddings. Since ψ is an S-immersion we have dimV = n − dim(ΠU).Then dimW ≥ dim(ΠU) + 1 + dimLk implies that dimV − 1 > dimLk.By lemma 8.5 the set V is open. Let U0 × C0 be a neighborhood of (φ, 0)inside radial−1(V) where C0 is a core. Then all ψ ∈ U0 are longitudinalS-embeddings at all y ∈ C0 ⊂ U ⊂W .

In order to extend to points y outside the local stratum we will useinduction. For each z ∈ Lk we may choose a tubular domain T z in Lkbased at z to get a depth k + 1 coordinate system uz. Inductively, we getcores Cz and neighborhoods Uz of φ such that all ψ ∈ Uz are longitudinalS-embeddings at y in the patch P z. There is a finite subcovering of Lkby {Czk+1 × [0, δzk+1] × Lz} indexed by z ∈ I. Define the core C∩ in u byC∩j =

⋂z∈I C

zj and δ∩j = minz∈I δ

zj for j ≤ k. Similarly let U∩ =

⋂z∈I Uz

define a neighborhood of φ. For all ψ in U∩ and y in the patch below thecore C∩, but outside the local stratum, we have that ψ is a longitudinalS-embedding at y.

Now for all ψ in the intersection U0∩U∩ we have that ψ is a longitudinalS-embedding at all y in the patch below C0 ∩ C∩. �

Proposition 8.7 Let W be a compact sharp stratified space. Let n >dimW . Assume that the sets of S-embeddings in Fro(L,Rm) are open forall links L occurring in W and m > dimL+ 1. Then the set of longitudinalS-embeddings in Fro(W,Rn) is open.

Proof: By lemma 8.6 above we choose for each x ∈ W a tubular do-main T x such that there is a neighborhood Ux of φ and a core Cx whereall ψ ∈ Ux are longitudinal S-embeddings at y in the patch. Since W iscompact there is a finite subcovering of {P x} indexed by x ∈ I. Then⋃x∈I Ux is a neighborhood of φ inside the set of longitudinal S-embeddings.

Consequently the set of longitudinal S-embeddings in Fro(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 dimW < n.Assume that the sets of S-embeddings in Fro(L,Rm) are open for all linksL occurring in W and m > dimL + 1. Given a depth k coordinate systemu there is a neighborhood U of φ and a core such that for each ψ ∈ U therestriction of ψ to the corresponding patch is injective.

30

Proof: By induction on the height, we may assume that theorem 8.9below already holds for sharp stratified spaces of lower height. Given W , wedefine the skeleton as the subspace W< of W where all strata of maximaldimension has been removed. The restriction induces a map Fro(W,Rn)→Fro(W<,Rn) and an S-embedding φ restricts to an S-embedding of W<.Consequently, by induction, there is an open neighborhood U< of mapsψ : W → Rn such that their restrictions to the skeleton are injective. Inorder to prove that such a map ψ ∈ U< is injective, it is enough to considerpoints 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 ofthe coordinate system. If u has maximal depth, then we may use the scaledmean value theorem 5.7. For a sufficiently small core and a neighborhoodU ⊆ 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 productspk−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, thenpk−1 6= 0. It follows that Λj(x, y) = 0, and consequently x = y. Hence allψ in some neighborhood U are injective on the patch corresponding to thecore chosen above.

To prove the inductive step we will establish several inequalities. Let usstate each such inequality carefully, and then prove it. Let pr t : Rn → Rnbe the orthogonal projection onto φ∗T

k0 Bu, i.e. the subspace of Rn spanned

the scaled Jacobi matrix Sukφ(0). Let Ht

kψ = pr tHkψ be the tangentialprojection of the Hadamard expansion. As usual V0 will denote the ortogonalcomplement of φ∗T

k0 Bu. Let prn be the projection onto V0, and let H⊥k ψ =

prnHkψ.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 andρ0, . . . , ρk for y. The weighted means of mixed products, p0, . . . , pk, are thengiven by the formula in Definition 5.5. Let r and ρ be the products

r = r0 · · · rk−1rk and

ρ = ρ0 · · · ρk−1ρk.

Since y lies in a top dimensional stratum, both ρ 6= 0 and pk−1 6= 0. Wewill omit the projection Bu → ΠU from our notation and write Hkψ(y) eventhough 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 ofφ such that for all ψ in U and points x, y in the core with pk−1 6= 0, we have∣∣Ht

kψ(x)−Htkψ(y)

∣∣ ≥ K1 |Λk(x, y)| .

Proof of claim 1: The function given by f(~v) = |pr tSukφ(0) ~v| has a

minimum value c on the set of unit vectors ~v. We can take c > 0 sincepr tS

ukφ(0) is injective. Since the limit of Mψ is Su

kφ(0) as ψ → φ andx, y → 0, there is a core and a neighborhood U of φ such that

|pr tMφ~v| ≥c

2for all x, y, ψ,~v with pk−1 6= 0.

Let ~v = Λk(x,y)|Λk(x,y)| and K1 = c

2 . Then

|pr tMφΛk(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 ofφ such that for all ψ in U and points x, y in the core with pk−1 6= 0, we have√

ρ2 + r2 ≥ K2

∣∣∣pr t(ρψ(y)− rψ(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 tφ(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 tψ(x)| for all x in the core.

Suppose r · ρ 6= 0. Take ~v1 = ψ(y), ~v2 = ψ(x) and let θ be given byρ =

√ρ2 + r2 cos θ, r =

√ρ2 + r2 sin θ. Furthermore set K2 = 1

c . Then theinequality √

ρ2 + r2 ≥ K2

∣∣∣pr t(ρψ(y)− rψ(x))∣∣∣

holds by a rearranging of the inequality above. In the case r · ρ = 0 theinequality is trivially true.

32

Claim 3: Let {T i} be a finite set of tubular domains in Lk together withcores Ci × [0, δi]× Li whose image cover Lk. Using that φ is a longitudinalS-embedding there exist a constant K3, a core of u and a neighborhood Uof φ such that for all ψ ∈ U and points x, y in the core of u, we have thereis an index i such that both x and y lie in the core

C0 × [0, δ0]× · · · × Ck × [0, δk]× Ci × [0, δi]× Li

or the following inequality holds,∣∣∣prn(ρψ(y)− rψ(x))∣∣∣ ≥ K3

√ρ2 + r2.

Proof of claim 3: For any pair of vectors ~v1 and ~v2 let β(~v1, ~v2) bedefined 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 thecore Ci × [0, δi] × Li of T i. Since φ is longitudinal S-embedding the radialmap at the basepoin φ⊥0 is ray injective, and there exists a number δ > 0such that

β(prnφ(x), prnφ(y)) ≥ δ

for (x, y) ∈ (L×L)rD. Choose a neighborhood U of φ and a core of u suchthat

β(prnψ(x), prnψ(y)) ≥ δ

2

for all ψ ∈ U and x, y inside the core such that the longitudinal projectionof (x, y) onto Lk × Lk does not lie in D. With K3 = δ

2 and θ as in claim2 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 ∈ Ci× [0, δi]×Li.This proves the claim.

Claim 4: Given any constant ε > 0 there exist a core and a neighborhoodU of φ such that for all ψ in U and points x, y in the core with pk−1 6= 0, wehave

ε |Λk(x, y)| ≥∣∣∣H⊥k ψ(x)−H⊥k ψ(y)

∣∣∣ .Proof of claim 4: Observe that the matrix prnMψ converges to 0 asψ goes to φ and x, y → 0. Given ε > 0 we may choose a core and aneighborhood U such that for all ψ ∈ U , x, y in the core with pk−1 6= 0 andall unit vectors ~u we have

ε ≥ |prnMψ · ~u| .

The inequality in the claim follows by the scaled mean value theorem bysetting ~u = Λk(x,y)

|Λk(x,y)| .

33

Proof of proposition: For each z ∈ Lk choose a tubular domain T z basedat z. This gives depth k+1 coordinate systems uz given by (T0, . . . , Tk, T

z).By induction, there are a core Cz of uz and a neighborhood Uz of φ insideU< such that each ψ in Uz is injective on the patch P z below the core Cz.Since Lk is compact, we may choose finitely many {T i} such that the coresCik+1× [0, δik+1]×Li cover Lk. Define a core C of u by taking the intersectionCj =

⋂Cij and δj = mini δ

ij . 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 con-

stants K1, K2 and K3 such that the inequalities of claim 1, 2 and 3 aresatisfied. Choose some ε < K1K2K3, and shrink the core and neighborhoodfurther 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 corewith pk−1 6= 0. We will show that x and y are identified in W . By theHadamard expansion

Hkψ(x)−Hkψ(y) = ρ0 · · · ρkψ(y)− r0 · · · rkψ(x).

The four inequalities now imply that

K1K2K3 |Λk(x, y)| ≤ K2K3

∣∣Htkψ(x)−Ht

kψ(y)∣∣

= K2K3

∣∣∣pr t(ρψ(y)− rψ(x))∣∣∣

≤ K3

√ρ2 + r2

≤∣∣∣prn(ρψ(y)− rψ(x))

∣∣∣=∣∣∣H⊥k ψ(x)−H⊥k ψ(y)

∣∣∣≤ ε |Λk(x, y)| .

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. Con-sequently the inequality of claim 3 does not hold. Therefore x and y liein the core of one of the deeper coordinate systems ui. By induction ψ isinjective on this deeper patch P i. Hence x = y. Thus all ψ in U are injectiveon the patch P of u. �

Theorem 8.9 Let W be a compact sharp stratified space and n > dimW .Then the set of S-embeddings is open in Fro(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 Cx and a neighborhood Ux such thatfor all ψ ∈ Ux we have injectivity of ψ restricted to the patch P x belowCx. By compactness of W there is a finite subcovering {P i} and a Lebesguenumber δ > 0 such that if |φ(x) − φ(y)| < δ then x, y ∈ P i for some patchfrom the finite subcoveing. Let U0 be the neighborhood of φ consisting ofall maps ψ such that |ψ(y)− φ(y)| < δ

2 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 thetriangle inequality

|φ(x)− φ(y)| ≤ |φ(x)− ψ(x)|+ |ψ(y)− φ(y)| < δ

2+δ

2= δ

it follows that x, y ∈ P i for some patch from the finite subcovering. Thenx = y since ψ ∈ U i is injective on P i. Consequently all such ψ are injectivelongitudinal S-embeddings. �

9 Existence of S-embeddings

In this section we will prove that S-embeddings exist into Rn when n issufficiently large. Furthermore the set of S-embeddings is dense among allsmooth maps. We will follow the approach found in [Hir76]. In order toprove 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 smoothand let g : Rn → R be a function. Consider the composition f = g ◦ φ. Fora tubular domain T we apply Hadamards lemma and find

f1

∣∣ρ0=0

= Dgφ0(x) · φ1

∣∣ρ0=0

.

Proof: Differentiating the equality f = f0 + ρ0f1 with respect to ρ0 andthen substituting ρ0 = 0 we get

∂f

∂ρ0

∣∣ρ0=0

= f1

∣∣ρ0=0

.

By the ordinary chain rule for g ◦ φ = g ◦ (φ0 + ρ0φ1) we get

∂

∂ρ0(g ◦ φ) = Dgφ(x) ·

(φ1 + ρ0

∂φ1

∂ρ0

).

By substituting ρ0 = 0 the result follows. �

35

Proposition 9.2 Let φ : W → Rn be an S-immersion. Then the followingare equivalent:

i) The map φ is a longitudinal S-embedding.

ii) For all depth k coordinate systems u the radial maps φ⊥0 : Lk → V k0

are ray injective at the basepoint.

Proof: Let us prove that i) and ii) are equivalent by induction on theheight 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 Nφ⊥0 , asin statement i). Let ψ = Nφ⊥0 : L→ V0. By the induction hypothesis, ψ be-ing a longitudinal S-embedding is equivalent to the following statement: Forall depth k−1 coordinate systems u of L the radial map ψ⊥0 : Lk−1 → V k−1

0

at the basepoint of u is ray injective. Here V0 is the orthogonal complementof ψ∗T

k−10 Bu.

Given such a depth k − 1 coordinate system u in L we construct thecoordinate system u in W by T0 = T and Tj = Tj−1 for j > 0. ThenLj = Lj−1. Now there are several subspaces of Rn to consider. From thetubular domain T there is the normal plane of the local stratum U at thebasepoint, which we denote V0. The scaled Jacobi matrix Su

kφ(0) spansφ∗T

k0 Bu, and V k

0 is its orthogonal complement in Rn. Note that V k0 ⊆ V0.

Using ψ and u defined on the link L we get the subspace ψ∗Tk−10 Bu in V0

spanned by S uk−1ψ(0). The complement in V0 is called V k−1

0 .We now want to compare the radial maps

φ⊥0 : Lk → V k0 and ψ⊥0 : Lk−1 → V k−1

0 .

We get φ⊥0 from the expansion

φ = φ0 + ρ0φ1 = φ0 + ρ0φ1 + . . .+ ρ0 · · · ρk−1φk

by the formula φ⊥0 = prV0 φk. Let Np be the composition of orthogonal

projection Rn → V0 and the normalization mapN(~v) = ~v|~v| in V0. We observe

that Np is defined for all ~v ∈ Rn not in the tangent plane of φ0 : U → Rnat the basepoint. With this notation we have

ψ = Np ◦ (φ1 + ρ1φ2 + . . .+ ρ1 · · · ρk−1φk).

36

By the chain rule we can compute the scaled Jacobi matrix

S uk−1ψ(0) = DNp

φ1(0) Suk−1φ1(0).

Furthermore, we get by the longitudinal chain rule that the map ψ⊥0 : Lk−1 →V k−1

0 is given as

ψ⊥0 = pr V k−10

DNpφ1(0)φk.

Since φ is an S-immersion it follows that the vector φ1(0), which is one ofthe columns in Su

kφ, is orthogonal to V k0 . By computation of the Jacobi

matrix of the normalization we have

pr V k−10

DNpφ1(0) =

1

|φ1(0)|prV k

0.

Hence

ψ⊥0 =1

|φ1(0)|incl ◦φ⊥0 ,

where incl is the inclusion of V k0 into V k−1

0 . By this formula it follows thatψ⊥0 is ray injective if and only if φ⊥0 is ray injective. This completes theinductive step. �

Lemma 9.3 If φ : W → Rn is an S-embedding and f : W → R a smoothfunction, then (φ, f) : W → Rn+1 is also an S-embedding.

Proof: Let u be a depth k coordinate system. For the scaled Jacobimatrices we have

Suk(φ, f) =

(SukφSukf

).

Since the column vectors of Sukφ are linearly independent, the larger matrix

also has this property. Moreover we have Vk = V ′k∩Rn, where Vk and V ′k arethe orthogonal complements of the column spaces of Su

kφ in Rn and Suk(φ, f)

in Rn+1 respectively. Consequently, prVkprRn = prVkprV ′k. This implies

thatprVk(φ, f)⊥k = prVkprV ′k

(φk, fk) = prVk φk = φ⊥k .

Suppose that y1, y2 in Lk satisfies

(φ, f)⊥k (y1) = λ(φ, f)⊥k (y2)

for some λ > 0. Apply prVk to transform this equation into

φ⊥k (y1) = prVk(φ, f)⊥k (y1) = λprVk(φ, f)⊥k (y2) = λφ⊥k (y2).

37

Then y1 = y2 since φ⊥k is ray-injective. It follows that (φ, f)⊥k also is rayinjective.

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 RNfor some large N .

Proof: By induction on the height of W . Manifolds can be embedded bythe well-known Whitney embedding theorem, see [Hir76, Theorem 1.3.5].

For the inductive step first consider a tubular domain T with link L andlocal stratum U ⊆ Rk. By induction choose an S-embedding ψ : L → Rn.Define φ : T → Rk+n+1 by

φ = φ0 + ρ0φ1,

where φ0 is the inclusion U ⊆ Rk ↪→ Rk+n+1 and φ1 is the composition

Lψ−→ Rn inv. stereo. proj.−−−−−−−−−−→ 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-embeddingφi : T i → Rni . Define ψi : W → Rni+1 by

ψi(x) = (gi(x), gi(x) · φi(x)).

Clearly each ψi is a longitudinal S-embedding at points where gi(x) > 0.Then by the lemma above we get that

(ψ1, ψ2, . . . , ψK) : W → RN , where N = n1 + n2 + . . .+ nK +K,

is everywhere a longitudinal S-embedding. However it is easily verified thatthis map is also injective. This constructs an S-embedding W → RN forsome large N . �

Given a unit vector ~u in Rn+1 we define the projection p~u : Rn+1 → Rnas the composition of orthogonal projection onto the normal space of ~u andthereafter the projection down to Rn ⊆ Rn+1. The last projection is bijectivewhen ~u /∈ Rn.

38

Proposition 9.5 Let W be a compact sharp stratified space of dimensionm. Let φ : W → Rn+1 be an S-embedding. If 2m + 1 ≤ n, then the set of~u ∈ RPn such that p~uφ : 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-manifoldhas measure 0 when m < n. We will construct finitely many subsets of RPnhaving measure 0 and such that for all ~u not in the union the compositionp~uφ is an S-embedding. First of all we want ~u /∈ RPn−1.

Suppose X 6= Y are strata of W . Let eX,Y : X × Y → RPn be thesmooth map sending (x, y) to the line with direction vector φ(x) − φ(y).Since dimX + dimY ≤ 2m < n the image EX,Y ⊆ RPn has measure 0.Observe that if ~u /∈ EX,Y ∪ RPn−1 then there is no intersection

p~uφ(x) = p~uφ(y), with x ∈ X and y ∈ Y .

Similarly for X a stratum consider eX : X×Xr∆→ RPn given by sending(x1, x2), with x1 6= x2, to the line with direction vector φ(x1)− φ(x2). Theimage EX ⊆ RPn has again measure 0. Define E to be the union

E =⋃X 6=Y

EX,Y ∪⋃X

EX .

Now whenever ~u /∈ E ∪ RPn−1, then p~uφ : W → Rn is injective.Next consider a max depth coordinate system u. Using the scaled Jacobi

matrix we define a map

fu : ΠU × RPm−1 → RPn

sending a point u ∈ ΠU and a direction vector ~v ∈ RPm−1 to the line withdirection vector Suφ(u) · ~v. The source is a manifold (with boundary) ofdimension 2m− 1. When 2m− 1 < n the image Fu of this map is a subsetin RPn of measure 0.

Cover W by finitely many max depth coordinate systems, and let F bethe union of the corresponding Fu. Observe that the composition p~uφ is anS-immersion whenever ~u /∈ F ∪ RPn−1.

At last consider a depth k coordinate system u with link Lk. We wantto construct subsets such that the composition p~uφ has ray injective radialmaps for all ~u not in these subsets. Let X 6= Y be strata in the link Lk. Wewill define a smooth

gu,X,Y : ΠU × (0,∞)×X × Y × Rl → RPn,

where l is the number of columns in Sukφ. This map sends (u, λ, z1, z2, ~v) to

the line with direction vector

φ(u, 0, z1)− λφ(u, 0, z2) + Suφ(u) · ~v.

39

To check that gu,X,Y is well defined, suppose that

φ(u, 0, z1)− λφ(u, 0, z2) + Suφ(u) · ~v = 0.

Then φ(u, 0, z1) − λφ(u, 0, z2) ∈ V ku , 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 RPn. Since dimW = m it follows

that l + 1 + dimX ≤ m, and similar for dimY . Hence

dim(

ΠU × (0,∞)×X × Y × Rl)≤ l + 1 + dimX + dimY + l < 2m.

Consequently, Gu,X,Y has measure 0 in RPn.Similarly, we may define a map

gu,X : ΠU × (0,∞)× (X ×X r ∆)× Rl → RPn,

using the same formula as above. Again the image Gu,X is a measure 0subset of RPn. Define Gu to be the union

Gu =⋃X 6=Y

Gu,X,Y ∪⋃X

Gu,X .

where X,Y run through all strata of Lk.Let ~u be a direction vector for some line in RPn r RPn−1 and suppose

that (p~uφ)⊥ is not ray injective at some u ∈ ΠU . Then there are pointsz1 6= z2 ∈ Lk and a positive number λ such that

(p~uφ)⊥(z1) = λ(p~uφ)⊥(z2).

Hence for some ~v ∈ Rl we have

p~uφ(u, 0, z1)− λp~uφ(u, 0, z2) + (Suk(p~uφ)(u)) · ~v = 0.

This implies that there exists a scalar r such that

φ(u, 0, z1)− λφ(u, 0, z2) + Sukφ(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 W by finitely many co-ordinate systems, we construct a set G =

⋃Gu of measure 0 in RPn.

We have shown above that all radial maps of p~uφ are ray injective when~u /∈ G ∪ RPn−1.

Summarizing, we see that p~uφ is an S-embedding when ~u lies outsidethe measure 0 set E ∪ F ∪G ∪ RPn−1. �

40

Theorem 9.6 Let W be a compact sharp stratified space of dimension m.The set of S-embeddings is dense in Fro(W,Rn) provided that 2m+ 1 ≤ n.

Proof: Let Un ⊆ RPn be the direction vectors represented by ~u /∈ Rn−1.Composing with the projections p~u we get a continuous map

p : Un ×Fro(W,Rn+1)→ Fro(W,Rn).

Let φ0 : W → Rn be an arbitrary smooth map. Consider an open neigh-borhood U of φ0 in Fro(W,Rn). Choose an S-embedding ψ : W → RN .Then (φ0, ψ) : W → Rn+N is also an S-embedding. By iterated composi-tion of the maps p above we get a map

Un × · · · × Un+N−1 ×Fro(W,Rn+N )→ Fro(W,Rn).

Observe that (~en+1, . . . , ~en+N , (φ0, ψ)) maps to φ0. By continuity there areneighborhoods Vj of ~ej+1 in Uj such that Vn × · · · × Vn+N−1 × {(φ0, ψ)}has image in U . Since the dimensions satisfy 2m + 1 ≤ n it follows byproposition 9.5 that each Vj contains a direction vector ~uj such that thecomposition

p~un · · · p~un+N−1(φ0, ψ)

is an S-embedding. This S-embedding lies in U . We have thus shown thatthe set of S-embeddings is dense in Fro(W,Rn). �

References

[Bat15] T. Augustin Batubenge, A Survey on Frolicher Spaces, Quaest.Math. 38 (2015), no. 6, 869–884. MR 3435960

[BC05] H. L. Bentley and Paul Cherenack, Boman’s theorem: strength-ened and applied, Quaest. Math. 28 (2005), no. 2, 145–178. MR2146387

[Bom67] Jan Boman, Differentiability of a function and of its composi-tions with functions of one variable, Math. Scand. 20 (1967),249–268. MR MR0237728 (38 #6009)

[BT09] A. Batubenge and H. Tshilombo, Topologies and smooth mapson initial and final objects in the category of Frolicher spaces,Demonstratio Math. 42 (2009), no. 3, 641–655. MR 2554959

[BT14] T. A. Batubenge and M. H. Tshilombo, Topologies on productand coproduct Frolicher spaces, Demonstr. Math. 47 (2014),no. 4, 1012–1024. MR 3290403

41

[Cap93] Andreas Cap, K-theory for convenient algebras, Ph.D. thesis,Universitat Wien, 1993, Thesis.

[Che00] Paul Cherenack, Frolicher versus differential spaces: a pre-lude to cosmology, Papers in honour of Bernhard Banaschewski(Cape Town, 1996), Kluwer Acad. Publ., Dordrecht, 2000,pp. 391–413. MR 1799765

[Che01] , Distributions via seminorms and Frolicher spaces,Steps in differential geometry (Debrecen, 2000), Inst. Math.Inform., Debrecen, 2001, pp. 85–94. MR 1859290

[DN07] Brett Dugmore and Patrice Pungu Ntumba, Cofibrations in thecategory of Frolicher spaces: Part I, Homology, Homotopy andApplications 9 (2007), no. 2, 413–444.

[FK88] Alfred Frolicher and Andreas Kriegl, Linear spaces and dif-ferentiation theory, Pure and Applied Mathematics (NewYork), John Wiley & Sons, Ltd., Chichester, 1988, A Wiley-Interscience Publication. MR 961256

[FM80] George K. Francis and Bernard Morin, Arnold Shapiro’s ev-ersion of the sphere, Math. Intelligencer 2 (1979/80), no. 4,200–203. MR 600227

[Fro82] Alfred Frolicher, Smooth structures, Category theory (Gum-mersbach, 1981), Lecture Notes in Math., vol. 962, Springer,Berlin-New York, 1982, pp. 69–81. MR 682945 (84g:58008)

[GTMW09] Søren Galatius, Ulrike Tillmann, Ib Madsen, and MichaelWeiss, The homotopy type of the cobordism category, ActaMath. 202 (2009), no. 2, 195–239. MR 2506750

[Ham82] Richard S. Hamilton, The inverse function theorem of Nash andMoser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65–222.MR 656198

[Hir76] Morris W. Hirsch, Differential topology, Springer-Verlag, NewYork-Heidelberg, 1976, Graduate Texts in Mathematics, No.33. MR 0448362

[HKM14] Thomas Hotz, Andreas Kreigl, and Peter W. Michor, Frolicherspaces as a setting for tree spaces and stratified spaces, Ober-wolfach Report, vol. 11, 2014, pp. 2505–2509.

[HW01] Bruce Hughes and Shmuel Weinberger, Surgery and stratifiedspaces, Surveys on surgery theory, Vol. 2, Ann. of Math. Stud.,vol. 149, Princeton Univ. Press, Princeton, NJ, 2001, pp. 319–352. MR 1818777

42

[KM97] Andreas Kriegl and Peter W. Michor, The convenient setting ofglobal analysis, Mathematical Surveys and Monographs, vol. 53,American Mathematical Society, Providence, RI, 1997. MRMR1471480 (98i:58015)

[Kre10] Matthias Kreck, Differential algebraic topology, Graduate Stud-ies in Mathematics, vol. 110, American Mathematical Society,Providence, RI, 2010, From stratifolds to exotic spheres. MR2641092

[LSZ81] F.W. Lawvere, S.H. Schanuel, and W.R. Zame, On C∞ functionspaces, Preprint, 1981.

[Luc02] Wolfgang Luck, A basic introduction to surgery theory, Topol-ogy of high-dimensional manifolds, No. 1, 2 (Trieste, 2001),ICTP Lect. Notes, vol. 9, Abdus Salam Int. Cent. Theoret.Phys., Trieste, 2002, pp. 1–224. MR 1937016

[Mat70] John Mather, Notes on topological stability, Notes, HarvardUniversity, July 1970.

[Mil56] John Milnor, On manifolds homeomorphic to the 7-sphere, Ann.of Math. (2) 64 (1956), 399–405. MR 18,498d

[Mil61] , Two complexes which are homeomorphic but combina-torially distinct, Ann. of Math. (2) 74 (1961), 575–590. MR0133127

[MW07] Ib Madsen and Michael Weiss, The stable moduli space of Rie-mann surfaces: Mumford’s conjecture, Ann. of Math. (2) 165(2007), no. 3, 843–941. MR MR2335797

[Qui88] Frank Quinn, Homotopically stratified sets, J. Amer. Math. Soc.1 (1988), no. 2, 441–499. MR 928266

[Sie72] L. C. Siebenmann, Deformation of homeomorphisms on strati-fied sets. I, II, Comment. Math. Helv. 47 (1972), 123–136; ibid.47 (1972), 137–163. MR 0319207

[Sik72] Roman Sikorski, Differential modules, Colloq. Math. 24(1971/72), 45–79. MR 0482794

[Sma58] Stephen Smale, A classification of immersions of the two-sphere, Trans. Amer. Math. Soc. 90 (1958), 281–290. MRMR0104227 (21 #2984)

[Sta11] Andrew Stacey, Comparative smootheology, Theory Appl.Categ. 25 (2011), No. 4, 64–117. MR 2805746 (2012g:18012)

43

[Sta13] , Yet more smooth mapping spaces and their smoothlylocal properties, http://arxiv.org/abs/1301.5493, January 2013.

[Tho69] R. Thom, Ensembles et morphismes stratifies, Bull. Amer.Math. Soc. 75 (1969), 240–284. MR 0239613

[Tho71] , Stratified sets and morphisms: local models, Proceed-ings of Liverpool Singularities–Symposium, 1 (1969/70), Lec-ture Notes in Mathematics, Vol. 192, Springer, Berlin, 1971,pp. 153–164. MR 0283808

[Tro07] David Trotman, Lectures on real stratification theory, Singular-ity theory, World Sci. Publ., Hackensack, NJ, 2007, pp. 139–155. MR 2342910

[Wal99] C. T. C. Wall, Surgery on compact manifolds, second ed., Math-ematical Surveys and Monographs, vol. 69, American Mathe-matical Society, Providence, RI, 1999, Edited and with a fore-word by A. A. Ranicki. MR 2000a:57089

[Whi65] Hassler Whitney, Tangents to an analytic variety, Ann. ofMath. (2) 81 (1965), 496–549. MR 0192520

44