45
TRANSACTIONS of the AMERICANMATHEMATICAL SOCIETY Volume 317, Number 2, February 1990 REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS: AN ELEMENTARY APPROACH H. J. SUSSMANN Abstract. We give a proof of a theorem on desingularization of real-analytic functions which is a weaker version of H. Hironaka's result, but has the ad- vantage of being completely self-contained and elementary, and not involving any machinery from algebraic geometry. We show that the basic facts about subanalytic sets can be proved from this result. 1. Introduction Subanalytic sets were studied by Gabriélov [G], Hironaka [Hil, Hi2], and Hardt [HI, H2, H3, H4], as a natural extension of the theory of semianalytic sets developed by Lojasiewicz [LI, L2]. The theory of subanalytic sets and their associated stratifications can be developed in two ways, namely (a) by appeal- ing to H. Hironaka's theorem on resolution of singularities, or (b) by means of a direct approach, as suggested in Hardt [HI]. The first approach makes it possible to prove stronger results, such as the fact that every subanalytic set is a locally finite union of images of open cubes under maps that are real analytic on a neighborhood of the closure of the cube. (In Hironaka's approach, this is actually how subanalytic sets are defined.) The second approach, however, has the advantage of not requiring the use of the formidable apparatus of desingu- larization theory. A detailed account of how this approach can be carried out to a successful conclusion will appear in [S]. The purpose of this note is to show that it is possible to obtain the full force of the results of the first approach by completely elementary means. We give a simple proof of a desingulariza- tion theorem which, although weaker than Hironaka's, suffices to obtain all the results on subanalytic sets. This "weak desingularization theorem" says that every nontrivial real-analytic function / on a real-analytic manifold M can be "desingularized" in the sense that there exists a real analytic, proper, surjective map <P: N —* M, where A is a Cw manifold of the same dimension as M, such that the composite map / o <p is locally a monomial (i.e. has "normal crossings"). This is weaker than the full desingularization theorem, in that it Received by the editors May 3, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 32C45; Secondary 32B20, 32C42. The author was partially supported by NSF Grant No. DMS83-01678-01. ©1990 American Mathematical Society 0002-9947/90 $1.00+ $.25 per page 417 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

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TRANSACTIONS of theAMERICAN MATHEMATICAL SOCIETYVolume 317, Number 2, February 1990

REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS:AN ELEMENTARY APPROACH

H. J. SUSSMANN

Abstract. We give a proof of a theorem on desingularization of real-analytic

functions which is a weaker version of H. Hironaka's result, but has the ad-

vantage of being completely self-contained and elementary, and not involving

any machinery from algebraic geometry. We show that the basic facts about

subanalytic sets can be proved from this result.

1. Introduction

Subanalytic sets were studied by Gabriélov [G], Hironaka [Hil, Hi2], and

Hardt [HI, H2, H3, H4], as a natural extension of the theory of semianalytic

sets developed by Lojasiewicz [LI, L2]. The theory of subanalytic sets and their

associated stratifications can be developed in two ways, namely (a) by appeal-

ing to H. Hironaka's theorem on resolution of singularities, or (b) by means

of a direct approach, as suggested in Hardt [HI]. The first approach makes it

possible to prove stronger results, such as the fact that every subanalytic set is

a locally finite union of images of open cubes under maps that are real analytic

on a neighborhood of the closure of the cube. (In Hironaka's approach, this is

actually how subanalytic sets are defined.) The second approach, however, has

the advantage of not requiring the use of the formidable apparatus of desingu-

larization theory. A detailed account of how this approach can be carried out

to a successful conclusion will appear in [S]. The purpose of this note is to show

that it is possible to obtain the full force of the results of the first approach

by completely elementary means. We give a simple proof of a desingulariza-

tion theorem which, although weaker than Hironaka's, suffices to obtain all the

results on subanalytic sets. This "weak desingularization theorem" says that

every nontrivial real-analytic function / on a real-analytic manifold M can be

"desingularized" in the sense that there exists a real analytic, proper, surjective

map <P: N —* M, where A is a Cw manifold of the same dimension as M,

such that the composite map / o <p is locally a monomial (i.e. has "normal

crossings"). This is weaker than the full desingularization theorem, in that it

Received by the editors May 3, 1987.

1980 Mathematics Subject Classification (1985 Revision). Primary 32C45; Secondary 32B20,

32C42.The author was partially supported by NSF Grant No. DMS83-01678-01.

©1990 American Mathematical Society

0002-9947/90 $1.00+ $.25 per page

417License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

418 H. J. SUSSMANN

does not give the extra conclusion that the map can be taken to be a diffeomor-

phism on an open dense set. However, this extra condition is not needed at all

to develop the theory of subanalytic sets, as we show in the last two sections

of the paper. In those sections we show that the weak desingularization result

suffices to get, by completely elementary means, the basic results on stratifica-

tions of subanalytic sets, plus the theorem that the complement of a subanalytic

set is subanalytic. We limit ourselves to the most basic stratification theorems,

namely, the stratifiability of subanalytic sets (Theorem 9.1) and maps (Theo-

rem 9.2). Theorem 9.2 gives the existence, for a Cw map /:M->JV which

is proper on a closed subanalytic set L Ç M, and locally finite families sf , AAAA!

of subanalytic sets of M ,N, respectively, of stratifications AT ,AT, by Cw ,

connected, embedded subanalytic submanifolds that are subanalytically diffeo-

morphic to balls, with the property that every A e si u {L} is a union of

strata of AT, every B e AA§ U {f(L)} is a union of strata of AT, and the re-

striction AT\L of AT to L is trivial over AT\f(L), in the sense that every

S eAT\L is subanalytically diffeomorphic to a product Cs x Ts by means of a

map <&s: Cs x Ts —» S, where Ts eAT, Cs is an open cube in R for some k ,

and /o$j is the canonical projection n: Cs x Ts —> Ts . Of course, these re-

sults are not new, but the proofs presented here are—we believe—considerably

clearer and simpler than others available in the literature. (Cf., e.g. the pa-

pers by Hardt [HI, H2, H3, H4]. Our method is essentially the "corank one

projections" method of Hardt [H4].)

Recently, E. Bierstone and P. Milman [BM] have independently found an-

other elementary proof of a weak desingularization result that also makes it

possible to carry out the development of subanalytic set theory exactly as is

done here.

2. Statement of the weak desingularization theorem

Throughout this paper, the word manifold means finite-dimensional, Haus-

dorff, second countable manifold of pure dimension (i.e. such that all the con-

nected components have the same dimension). All manifolds considered will be

of class Cw, i.e. real analytic. If M,N are Cw manifolds, then CW(M,N)

denotes the set of all Cw maps from M to N. The elements of CW(M,R)

are the Cw functions on M, and those of CW(M ,C) are the complex-valued

C0J functions on M . A function /: M —► R is nontrivial if it does not vanish

identically on any connected component of M. (If / is Cw , it follows that

{x:x e M, f(x) ^ 0} is open and dense in MA) A C map f:M —> N has

full rank if every connected component of M contains at least one point p

where the differential of / has rank equal to v = min(dimM, dim A). (If

f e COJ(M ,N), it then follows that df has rank v on an open, dense subset

of MA) A continuous map /: M —► N is proper if f~ (K) is compact in M

for every compact subset K of N .

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REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS 419

We use C"(e) to denote the open cube

(2.1.a) C"(e) = {(xx, ... ,xn) e R": |x(| < e for i = 1, ... ,n},

and write

(2.1.b) C"c(e) = {(z, ,...,zn)eC": \z,\ < e for i = I, ... ,n}

for the corresponding complex polydisc. (Sometimes we will write C^(e) for

C"(e), to emphasize that we are dealing with a real cube rather than a polydisc.)

If M is a Cw manifold, and dim M = n, a cubic chart of M is a triple

(U,<$,e) where

(2.1.a) U is an open subset ofM,

(2.1.b) <P: U —» C"(e) is a Cw diffeomorphism.

The point p = <P~'(0) is the center of the chart (C/,<P,e), the number s is

its radius, and the set £/ is its domain.

We let Z+ denote the set of nonnegative integers (so that 0 e Z+). If

a = (a.a„)6Z" and { = (£, ,...,<*„) €C" , we let

(2.2.a) r=CC -Cand

(2.2.b) |a| = c^ + a2-I-hQn.

A Cw monomial with respect to a cubic chart (£/,<P,e) isa Cw function

/: LA —> C such that there is a nowhere vanishing Cw function A:®(U) —* C

and a multi-index aeZ", such that

(2.3) f(<t>~fx)) = A(x)xa forxe<&(U).

(Clearly, / is real valued iff A is.)

A function / e CW(M, R) is desingularized if every p e M is the center of

a Cw cubic chart (C/,<P,e) such that the restriction of / to U is a monomial

with respect to (t/,<P,e). More generally, if AT is a set of Cw functions on

M, we call AT desingularized if every p e M is the center of a Cw cubic chart

(c7,<P,e) such that all the restrictions to U of the members of AT are mono-

mials with respect to ( U, <P, e). (Notice that this is stronger than requiring that

each feAT be desingularized.)

Let M be a Cw manifold, and let .T be a subset of CW(M,R). A desin-

gularization of AT is a pair (N ,v) such that

(2. II.a) N is a Cw manifold of the same dimension as M,

(2.II.b) v: N —> M is a proper, surjective Cmmap,

(2.II.C) the set AT o v = {fov.fe AT} is desingularized.

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420 H. J. SUSSMANN

If A is a Cw manifold, we use <S(N) to denote the set of connected com-

ponents of N. We emphasize that it is part of the definition of a manifold that

all the Q e ¿f(A) have the same dimension. A union of tori is a pair (N ,9),

where A is a Cw manifold, and 9 = {9Q:Q e ¿f(A)} is a family of Cw

diffeomorphisms 9 : Q —> T" . Here « = dimA and Tn is the «-dimensional

torus (S )" . The diffeomorphism 9 is given by an «-tuple (9X, ... ,9^) of

Cw functions from Q to S , that will be referred to as a system of angular

coordinates for Q . If (N ,9) is a union of tori, then a function /: A —> R will

be said to be a sine monomial on (A, 9) if, for each Q e S(N), the restriction

f\Q of f to Q is given, in terms of the coordinates (9°, ... , 9°), by

(2.4) f(9? ,...,9Qn) = AQ(9f,..., 0„G)[sin 9Qf

for some multi-index a and some nowhere vanishing Cw function A : Q —*

R. (Here sin0e denotes the «-tuple (sinOp, ... ,9®).)

We let Mon( A, 9) denote the set of all sine monomials on (N ,9). It is easy

to see that Mon(A,0) is desingularized.

If AT ç Cœ(M,R), a toricdesingularization of AT is a triple (N,9,v) such

that

(2.III.a) A is a union of tori, and dim A = dim M,

(2.III.b) v: N —► M is a proper, surjective Cwmap,

.. IIT , the set AT o v = {f o v: f e AT} consists of sine monomials on(2.III.C) {NJ)

It is clear that, if (N,9,v) is a toric desingularization of AT, then (A,v)

is a desingularization.

Our weak desingularization theorem says that every nontrivial fe CW(M, R)

has a desingularization. The proof will actually construct a toric desingulariza-

tion, and it will be useful in applications to have this stronger property, so we

will incorporate it in our statement. Moreover, it will be convenient to add an

extra property that follows trivially from the others. Suppose that (A ,v) is a

desingularization of AT ç CW(M ,R). Let À be the union of those connected

components Q of A such that v has rank equal to dim M at some point of

Q, and let v be the restriction of v to Ñ. Then v(N) is a closed subset

of M. Since v is surjective and v(N - Ñ) is nowhere dense, it follows that

v(Ñ) = M . So (A, v) is also a desingularization of AT . Hence we can assume

that v has full rank. Finally, it is easy to see that, if every function has a desin-

gularization, then the same is true for every finite set of functions (cf. Lemma

3.4 below). We prefer to incorporate this in our statement. So, the theorem to

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real-analytic desingularization and SUBANALYTIC SETS 421

be proved is

Theorem 2.1. Let M be a real-analytic manifold, and let AT be a finite set

of nontrivial real-valued real-analytic functions on M. Then AT has a toric

desingularization (N ,9 ,v) suchthat v has full rank.

We will refer to Theorem 2.1 as the "Weak Desingularization Theorem". Hi-

ronaka's desingularization theorem is much stronger, for it asserts the existence

of a desingularization (N,v) such that v is a diffeomorphism on an open

dense set.

3. From germs to functions

The proof of Theorem 2.1 will be by induction on « = dim M. We remark

that the cases « = 0 and « = 1 are trivial. From now on we fix an «, and

assume that Theorem 2.1 is true for all manifolds of dimension < « . Our goal

is to prove the theorem when dim M = « + 1 . We will do this by proving

the desired conclusion for germs. This requires that we define what is meant

by a desingularization of a germ. The purpose of this section is to give this

definition, and to prove that the desingularizability of germs implies that of

functions.

If v: A —> M is a continuous map, and A is a compact subset of A, we say

that v is open at K if, whenever U is a neighborhood of K, it follows that

v(U) contains a neighborhood of v(K).

If v:N -* M is continuous, and q e N, then every germ f of functions at

v(q) has a well-defined pullback v*(f), which is a germ at q.

If M is a Cw manifold, and p e M, we let M(f denote the set of all

germs at p of Cw real-valued functions defined on neighborhoods of p. If

v:N —> M is a Cw map, and q e N, then v* maps M<fv, . to N(f .

If f eM (A? , we say that f is desingularized if there exists a C0J cubic chart

(£/,<P,e), centered at p , and a representative / of f, which is defined on U

and is a Cw monomial with respect to ( LA, <P, e). More generally, let A§ çM <f

be a set of germs at p of Cw functions. We say that AAA is desingularized if there

exists a Cu cubic chart (C7 ,<P,e), centered at p , such that every f e & has a

representative f e Cw(c7,R) which is monomial with respect to (c7,<P,e).

A desingularization of a set "§ of germs at p of C0J functions on M is a

triple (N,K,v) suchthat

(3.1.a) A is a Cw manifold, of the same dimension as M,

(3.1.b) v is a Cwmap from M to N,

(3.1.c) K is a compact subset N,

(3.1.d) HK) = {p},

(3.1.e) v is open at K,

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422 H. J. SUSSMANN

(3 I f) f°r eVery Q € K ' tke S6t °fgermS V*q^) = {v*q(f)'-f S ^} is

desingularized.

The set of germs 3? is desingularizable if it has a desingularization.

If / e CW(M,R), and p e M, we let fp denote the germ of / at p. If

9" ç CW(M, R), we let AT = {fp: feAT}.We then have

Lemma 3.1. Let M be a Cw manifold and let AT c CW(M,R) be finite. Then

AT has a desingularization if and only if AT is desingularizable for every p e M.

In that case, AT has a toric desingularization (N ,9 ,v) such that v has full

rank.

To prove Lemma 3.1, we first make a trivial observation and prove another

lemma.

Observation 3.2. If N,Q are locally compact metric spaces, and v:N —> Q is

continuous, surjective and proper, then v is open at v~ (K) for every compact

subset K of Q. o

Lemma 3.3. Let M be a Cw manifold, p e M, AÀAA çM & Assume 2> is

finite. Then the following are equivalent:

(i) & is desingularizable,

(ii) there exist N,v,U,AT such that: (a) A is a compact Cw manifold,

(b) dim A = dim M, (c) v.N^M isa Cw map, (d) U is a neighborhood of

p, (e) v(N) ç U, (f) u(N) contains a neighborhood of p, (g) AT ç CW(U ,K),

(h) ATp= g?, and (k) AT o v is desingularized,

(iii) every neighborhood V of p contains a U such that there are N ,9 ,v ,AT

for which (a),... , (h) of (ii) holds and, in addition, (N ,9) is a union of tori

and AT ov C Mon(N,9).

Proof. It is clear that (iii) implies (ii). To see that (ii) implies (i), let N,v , U,

AT be as in (ii), let g = v(N), K = {p}, J = v~\K) . Observation 3.2 tells us

that, if W is a neighborhood of / in A, then v(W) contains a neighborhood

Z of p relative to v(N). Since p is an interior point of v(N), Z also

contains a neighborhood of p in M. So v is open at J. Then (N,J,v) is

a desingularization of &.

We now prove that (i) implies (iii). Let (N,K,v) be a desingularization of

A§. Let F be a neighborhood of p. Pick a neighborhood U of p which is

connected, is a subset of V , and is such that every f e A§ has a representative

which is defined and real analytic on U. This representative is then unique.

Let AT be the set of all these representatives. For each q e K, pick a cubic

chart (W ,<P ,£ ), centered at q, such that every member of v*(&) has a

representative which is defined on W and is a Cw monomial with respect to

( W' <P , g ). By making e smaller, if necessary, we may also assume that

v(W ) ç U . If f e AT, then f o v is defined and analytic on W . Moreover,

(f ov) = v*(f ). Therefore the restriction <j>- of f ov to W is the unique

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real-analytic desingularization and subanalytic sets 423

real-analytic representative of v* (f ) whose domain is W . It follows that tp

is a monomial with respect to (W ,& ,e ). Choose qx, ... ,qm such that

(3.1) AçU<(C"(t))=Q-

Let TV,, ... ,Ñm be disjoint copies of the torus T", and let 9J:Ñ. —► T" be

Cw diffeomorphisms, given by 9J = (9\, ... ,9Jn). Let A be the union of the

Nj, and let 9 = {9J:j = 1, ... ,m}. Then A is compact, and (N,9) is a

union of tori. Define a map p: A —> A by letting p = p. on A , where p is

given in coordinates, by

^(sinÖ1)...,sinöi)(3.2) Pj(e{,...,djn) = <t>tlij

Then p e CW(Ñ ,N), and p(Ñ) contains the open set Q. Let v = v o p .

Then A is compact, dimA = dimAf, v e Cœ(N, M), U is a neighborhood

of p , v(N) ç U, v(N) contains a neighborhood of p (because v is open at

K), AT ÇCW(U,R), 9p = êA, (Ñ,9) is a union of tori and 9 o v consists

of sine monomials on ( A, 9). D

We now prove Lemma 3.1. Suppose AT is desingularizable. Let (N,v)

be a desingularization of AT. Let p e M. Let K = v~ (p). Since v is

proper and v(N) = M, Observation 3.2 implies that v is open at A. If

q e K, then q is the center of a cubic chart ( W , <P , e ), such that f o v is

a monomial with respect to ( Wt' , O , £ ) for each f e AT . If f e AT then

f = fp for some f e AT. Then (/o v)q = z/(f). So all the v*q(f), ieATp,

have representatives which are monomials with respect to (W ,<P ,e ). So

(N,K,v) is a desingularization of AT . This completes the proof of one of

the implications. To prove the other one, let AT ç CW(M ,R) be such that AT

has a desingularization for each p. Find a sequence (Kx ,JX), (K2,J2), ...

of pairs of compact subsets of M, such that \J*LX K. = M, AC IntJf

and {Jf.j = 1, ...} is locally finite. For each K., and each p e K., use the

equivalence (i) o (iii) of Lemma 2.4 to find (Nj p,9 j , v} ) such that A is

a compact manifold, dim A = dimAf, v• „:A- -> M is Cw , v}, p(N. )

contains a neighborhood of p, ^, „(A ) ç Int/ , (N- ,6J) is a union of

tori, and AT ov. ç Mon(A , Ö7). For each 7', pick a finite set ^ of points

p e #,, such that KjÇ\JpePjVj,p(Nj,p).

Let A be the disjoint sum of all the A. , for 7' e {1,2, ...} and /? e

^. If q e N, let 7, p e ATj be such that q e A. p. Then let z^(^) =

^z „(<?) • The map v:N ^ M is Cw, and surjective. If L ç Af is compact,

then L only meets finitely many J . But L can only intersect zv(A ) if

L (l Jj t¿ 0. So L only intersects finitely many sets v(Njp),peAT.. So

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424 H. J. SUSSMANN

v (L) is contained in the union of finitely many A , p e AT.. Therefore

v~ (L) is compact. So v is proper. If 9 = {9. :j = 1,2, ... ;p e ATA),

then (A ,9) is a union of tori. Finally, it is clear that ATov ç Mon(A,ö).

Therefore (A, 9, v) is a toric desingularization of AT. To conclude the proof,

we remark that, as indicated before the statement of Theorem 2.1, whenever

a toric desingularization (A,9,v) exists, then it is possible to eliminate the

connected components of A where v does not have full rank, and obtain a

toric desingularization with full rank. D

Next we show that, in order to desingularize a finite set of germs at a point,

it suffices to be able to desingularize a single germ. The relevant fact is the

following trivial lemma.

Lemma 3.4. Let fx ,f2 be Cw functions on a cube Cr(e), such that fxf2 is a

monomial. Then fx and f2 are monomials, a

Lemma 3.4 implies a similar conclusion for germs. In particular, it follows

that, in order to desingularize a finite set of germs at p, it suffices to desingular-

ize their product. Hence, in order to prove Theorem 2.1, it suffices to show that

every germ of a nontrivial Cw function on an (« + l)-dimensional manifold

M can be desingularized, assuming that Theorem 2.1 holds for manifolds of

dimension < « .

Our next lemma will play a crucial role in the proof of Theorem 2.1, since

it will enable us to show that a germ f is desingularizable provided that there

are pairs v , A such that the pullbacks i/*(f) are, in some sense, "simpler" than

f. If the "simplicity" is measured by some integer k(T) , in such a way that

k(í) = 0 implies that f is desingularizable, then Lemma 3.5 implies that every

f is desingularizable, provided that for every f we can find v such that all the

pullbacks v*(f) satisfy k(v*(ï))<k(ï).

Lemma 3.5. Let M be a Cw manifold, peM, feM(?p. Let (N,K,v) be

such that (3.1.a,b,c,d,e) hold, and

(3.1.^) for every q e K , the germ u*(f) is desingularizable.

Then f is desingularizable.

Proof. Using Lemma 3.3 pick, for each q e K, a compact Cw manifold

A , whose dimension equals dimN, and a Cw map v : A —* A, such that:

(i) v (N ) contains a neighborhood of q, (ii) v (N ) ç U , where Uq is a

neighborhood of q such that U ç v~ (V), where F is a fixed neighborhood

of p on which a representative / of f is defined, (iii) U is the domain of a

representative tpq of v*(f), (iv) U is connected, and (v) tpq o vq is desingu-

larized.

Since f o (v\U ) is also a representative of v*(f), it follows that fo (v\U )

and tpq coincide on a neighborhood of q, and therefore on all of U , since

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real-analytic desingularization and subanalytic sets 425

U is connected. Therefore, the function r -* f(v(r)), from A to M,

is desingularized. Let ¿f be a finite set of points q e K such that K ç

U €/gV.(N). Let A be the disjoint union of the A , and let v:N —► A

be given by v(r) = v (r) if r e A , q e(S. Then v(N) is a neighborhood

of K, and so v(v(N)) is a neighborhood of p, because v is open at K. Let

p:N —► M, p = v ov. Then p is Cw, A is compact, p(N) contains a

neighborhood of p , p(N) ç V , and /o// is desingularized. By Lemma 3.3, f

is desingularizable. D

4. Reduction to the case of a monomial discriminant

As indicated before, our proof of Theorem 2.1 is by induction on « , and the

cases « = 0, « = 1 are trivial. We assume, from now on, that the conclusion of

the theorem is true for some « , and we try to prove it for « + 1 . As remarked

in §3, it is sufficient to prove that every (« + 1 )-dimensional real-analytic germ

f is desingularizable. Clearly, it suffices to assume that M = Rn+ and f is

a germ at 0. After making a linear change of coordinates, we may assume,

by the Weierstrass Preparation Theorem, that f = u • p, where u is such that

u(0) / 0, and p is a distinguished polynomial. If we write R"+ as R" x R,

and use x = (xx, ... ,xn) for the coordinates of R" , and t for the coordinate

of R, we may assume that p has a representative p, defined on Cn+ (e), given

bym

(4.1) p(x,t) = tm + Y,PM)tm~fi=i

where px, ... ,pm are Cw real-valued functions on C"(e), such that px(0) =

• • • = Pm(0) = 0. Also, we may assume that u has a representative u that is de-

fined on Cn+ (e) and never vanishes. If p is desingularizable, then Lemma 3.3

gives us the existence of (N,v) with A compact, v:N —> Cn+ (e), v(N) 3 V ,

V a neighborhood of 0, and p desingularized. Then (up)ov is desingularized,

and therefore f is desingularizable. Therefore, in order to prove the desingu-

larizability of f, it suffices to prove that p is desingularizable.

Let tf" denote the ring of germs at 0 of real analytic functions on neighbor-

hoods of 0 e R" , and let J?" denote the field of quotients of ¿f " . We can

regard p as a member of .ATAn\t\, the ring of polynomials in t with coefficients

in A%An. In fact, we have

m

(4.2) p = t +2^P,ii=i

where p,, ... ,pm are the germs atO of px, ... ,pm . Let g denote the greatest

common divisor of p and p' (where the prime denotes derivative with respect

to /). Then we can write

(4.3) P = a-g

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426 H. J. SUSSMANN

where a e ATAn\f\. Of course, g is only defined up to multiplication by a

nonzero member of 3AAn . However, since AAAn is a unique factorization domain,

Gauss' Lemma implies that we can take a and g to have coefficients in cfn .

Moreover, since p is monic, we can take a and g to be monic as well.

If 3ÍA denotes an algebraic closure of ATAn , then p can be expressed as

(4.4) p(o=nc-¿,f1=1

where Xx, ... ,X are the distinct roots of p, and ax, ... ,a are their multi-

plicities. Then a = Vf^ft-X^ . Therefore, if a denotes the largest of the at,

we have a factorization

(4.5) aa = bp,

where b e ATAn\t\. Again, by Gauss' Lemma, b has coefficients in cfn , and is

monic. If we write

(4.6) a = ;* + X>/-',i=i

(4.7) b = í' + ¿b/-\i=i

m—ß

(4.8) g = r"" + £ g/-"-',/=i

and pick representatives a¡, bi, gi of the a(, b(, g(, we see that the polynomials

'" + £,"=. a,(°)^_' - t""" + Er=T" 1,(0)«""""' divide tm , which shows that

the £1,(0) and gfO) vanish, so that a and g are distinguished polynomials.

Similarly, b is a distinguished polynomial. Since a has simple roots in Jf ,

it follows that the discriminant of a does not vanish (as an element of <f").

It is clear that, if we desingularize a, then aCT will be desingularized as well.

In view of Lemma 3.4, p will also be desingularized.

Thus we have established that, in order to prove our conclusion in general, it

suffices to prove that every germ p of a distinguished polynomial with a nonzero

discriminant is desingularizable. Moreover, it will be convenient in our proof

to deal with a polynomial p that has zero as a root. So, if pm ^ 0, we multiply

p by t. Clearly, if we show that Zp is desingularizable, Lemma 3.4 shows that

p is.

We now consider a germ at 0, of the form

m-\

(4.9) p = t + 2^ p¡t ,i=i

where the p( have representatives p¡, that are defined on C(e), and satisfy

p (0) = 0. And we assume that the discriminant A of p is a nonzero element

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real-analytic desingularization and subanalytic sets 427

of ¿f" . For x e C"(e), let A(x) denote the discriminant of the polynomial

t -* tm + E™7'/>;(*)im~' • Then A is the germ of A at 0. So the function A

does not vanish identically on C(e).

We now apply the inductive hypothesis, and desingularize A. Let (N,v)

be a desingularization of A. Let K = ^~'(0). Let Ñ = Nx]- e,e[ and let

v:Ñ —* Cn+l(e) be the map (q,t) -» (v(q),t). Then v is a C0J map from

Ñ onto C"+1(£), and v is proper. Let K = v~[(0) (i.e., K = Kx {0}).

By Observation 2.3, v is open at K. Suppose we prove, for every q e K,

that the germ v~(p) is desingularizable. Then Lemma 3.5 implies that p is

desingularizable. Now, if q e K, then q = (q, 0), q e K, and we can choose

a cubic chart (U, <P,3), centered at q, such that A o <p~ = A is a product

Ax" , where A e Cw(Cn(ô), R), A never vanishes, and a e Z" . If we let

pi(x) =pi(v(<&~ (x))), p (x,t) = tm+Yf?Jx Pj(x)tm~', then p is a function

on Cn+ (S) which is a representative of £~(p). So we have achieved a further

reduction. We may drop the # superscript (and relabel S as £) and assume

that:

(4.1.a) p is a Cw function on Cn+ (e), of the form

m-\

(4.10) p(x,t) = tm + YdPlWtm~fi=i

(4.1.b) the discriminant A(x) of the polynomial t—>p(x,t) is given,

on C(s) , by

(4.11) A(x) = A(x)xa

where A e Cw(C"(e),R), A never vanishes on C"(e), and aeZ"+.

Moreover, if (4.1.a, b) hold, then it is clear that we may make £ smaller, if

necessary, and also assume:

(4.1.c) the functions pt , A extend to bounded complex holomorphic

functions p{ , A : C¿(e) -+ C , and

(4.1.d) A never vanishes on Cc(e)

(Recall that C¡¡,(e) is the complex polydisc defined by (2.Lb).)

Clearly, if (4.1.a,b,c,d) hold, then p has a holomorphic extension p to the

complex polydisc C^f (e), and p is given by

m-\

(4.12) p (z, w) = wm + y^ p. (z)wm~l

(=1

for z e Cç(fi), \w\ < £ .

Finally, the function

(4.13) AC(z) = AC(z)za

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428 H. J. SUSSMANN

(where z" = zf ■ ■ ■ z'*") is clearly the discriminant of p , regarded as a poly-

nomial in w.

5. Global root functions

We assume that (4.1.a,b,c,d) hold, and we let pc , Ac be the functions defined

by (4.12), (4.13). For each /, let

(5.1) Hl(e) = {(zx,...,zn)eCnc(e):zi = 0).

Let

(5.2) jrn(e) = \jH?(e)i=i

and

(5.3) £2B(e) = C£(fi)-/*""(£).

Then Q"(e) is an open, connected, dense subset of C£(e). If z e fi"(e),c c

then the discriminant A (z) of the polynomial p (z,-) does not vanish, and

so p (z,-) has m distinct complex roots in C (one of which is 0, since the

summation in (4.12) does not contain a term of degree 0 in w). Moreover,

if z e Q"(fi), then there exist, in some neighborhood U of z (such that

U ç Çf(e)), m holomorphic functions z —► wfz), i = I, ... ,m , such that

p (z, wfz)) — 0 for z e U, i = I, ... ,m , and wfz) / wfz) for z e U,

i ^ j. If y:[0,1] —> fi"(e) is a continuous curve such that y(0) = y (I) = z,

then one can define continuous functions £,: [0,1] —> C such that Ç,(0) = wfz)

and p (y(t),Cft)) = 0 for 0 < t < 1 . The functions C, are unique. Each

number Ç(.(l) is a root of p (z,-). Hence y induces a permutation n of the

set {I, ... ,m}, such that Çfl) = w^fz).

Let k be an even integer > 0 such that

(5.4) ny = identity

for all curves y. (For instance, we may take k = 2 if m = 1 , and k = ml if

m > 1.) Let S = e' , and let F:C¡¡.(o) —> C£(e) be the map

(5.5) F«,,...,£,,) = ({?,...,<£).

Then F is holomorphic and onto. Moreover, F is proper. If we let F denote

the restriction of F to Cl"(ô), then F maps il"(ö) onto Q"(fi), and F is a

covering map, the cardinality of each fiber being k" .

Let { = (Í1,...,{J,)€C¿(¿). We define />f({), for i = 1, ... ,m - 1, by

pf(Ç) = pf(F(i)). For weC, we define

m-l

(5.6) pC(i,w) = wm + JftPl(Z)™m~i-i=i

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REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS 429

Also, we let Ac(i) = AC(F(Ç)), Ác(¿;) = AC(F(£)), i.e.

(5.7) ÁC(c¡) = ÂC(Ç)Çka.

Naturally, A (£) is the discriminant of the polynomial p (¿,, ■).

If we now take a £ e il"(S), then there exist m holomorphic functions

wx,...,wm on some neighborhood U of F(Ç), such that wfz) ^ wfz)

whenever z e U, i ¿ j and pc(z, wfz)) = 0 for all i e{l, ... ,m}, z e U.

If we let w¡ = w( o F, then wx, ... ,w are m holomorphic functions on

a neighborhood V of £, such that wf£,) ± w.(Ç) for i ^ j, £, e V, and

Pc(t,wft)) = 0 for i=l,...,m, ÇeV.

We now fix a Ç e£l"(S), and a connected neighborhood F of ¿J on which

a set W of m functions w. as above exists. We choose a particular way of

labelling the functions in W as wx, ... , wm , and keep it fixed from now on.

If ¡C is an arbitrary point in Çl"(ô), then we can choose a curve y: [0,1]-+

Q"(ô) such that y(0) = Ç, y (I) = ¿j*. We can then continue the functions w¡

along y, and define functions Ç; = [0,1] —► C such that ÇfO) = wf£¡) and, if7 e [0,1], then there is a neighborhood L7 of y (7) and a holomorphic function

îîj(:C7 - C such that pc(£, ,wft,)) = 0 for £ € Í7, and iZ>;(y(/)) = C,(0 for /

in a neighborhood of 7. We claim that

the ordered sequence (Çx(l), ... ,£„(!)) ¿focs not depend on which

curve y from Ç to C is chosen.

To see this, first observe that every such curve y is homotopic (with fixed

endpoints) to a curve y which is the concatenation of « curves yx, ... ,yn,

that satisfy:

y¡ isacurvein iï"(â) along which only the coordinate £,t varies,

while £j, ... ,C¡_X , ¿i;+I, ... ,£„ remain constant.

Hence it suffices to prove that, if £,,... ,£B are complex numbers in {z: 0 <

|z|<r5}, wx, ... ,wm are holomorphic functions on a neighborhood Í7 of

cf = (if, ...,{J), such that tfc,.(£) / wfd) for i¿j, ÇeU, pc(t;,wfÇ)) =

0, A:[0,1]-» {z:0< |z| < ¿} is a curve such that X(0) = <?, , and X*:[0,l]^

C£{S) is given by X*(t) = ({J, ... ^ ,A(í),íf+1, • • • ,íj), then the numbers

C,(l), obtained by analytically continuing the functions w¡ along X*, depend

only on X(l), but not on how X is chosen. To see this, it suffices to show that,

for each fixed ¿¡, , ... ,<fj_x , tf+x, ... ,<*° , the functions

^^wfi0x,...,^_x,^,t;lx,...,ct0n),

that are clearly well defined locally on {z:0 < \z\ < 3} , can actually be defined

globally. And this will follow if we show that, if ß: [0,1] —> {z:0 < \z\ < S}

is a circle, centered at 0, and oriented counterclockwise, ß* is defined like X*

above, and «: [0,1] —» C is a continuous function such that p(ß*(t), t](t)) = 0

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430 H. J. SUSSMANN

for t e [0,1], then n(l) = «(0). But, if ß is such a circle, then F o ß* is

a curve which is obtained from some closed curve y by iterating y k times.

If we continue the m roots of p(F(ß*(0)),w) = 0 along y, then we obtain

a permutation % of the roots. Continuation along F o ß* therefore induces

the permutation ny , i.e. the identity. Therefore «(1) = «(0), as stated. This

completes the proof of (5.1).

In view of (5.1), we have m globally defined holomorphic functions wx, ... ,

wm on Q"(ô), such that wf£) / wft,) whenever i ^ j, £ e £l"(ô), and

pc(tA\,wfti)) = 0 for £, e n"(S), i = 1, ... ,m. Since the pf are bounded

(by (4.1.c)), it is clear that the p( are bounded. But then the wt are bounded

as well. So the w¡ extend to complex holomorphic functions on the whole

polydisc Cç(ô). From now on, we use Wj to denote the extended functions.

If ¿i e Çl"(ô), then the polynomial p (¿;, •) has m distinct roots wx(Ç), ... ,

wm(£,). Since p (£,,-) is monic, we have the identity

m

(5.8) pC(tl,w) = Y[(w-wfQ).i=i

Both sides are holomorphic functions of ¿;, w in C^(ô) x C. Therefore identity

(5.8) holds on C[l(ô)xC.

We now let S denote the set of all sequences a = (ax, ... ,an), at. = 1 or

er. = -1 , /= 1,...,« . Then S has 2" elements. For each a , we can define

a map <PCT: C^(ô)x] -£,£[—> C^+ (e) by letting

(5.9) tf(yx,...,yn,t) = (<Jxykx,...,onykn,t).

The maps <PCT are proper and of class Cw . Each <PCT maps Cr[(S)x] - e,s[

onto Cg'CT(fi)x]-£,£], where C^'a(e) is the "orthant"

{(xx, ... ,xn): 0 < oixl < e for /' = 1, ... , «}.

(Recall that k is even.)

Let M be the disjoint union of 2" copies Ma of C^(S)x] - e,e[, one for

each er e S. Define a map <P: M —► C^+ (e) by letting O = O'7 on Ma . Then

O is proper and surjective. Moreover, <P~ (0) consists of 2" points O^rreS,

where 0a is the origin of Ma , when ma is identified with C^(S)x] -£,£[.

Each map <Pa factors via G, where G is the map (Çx, ... ,Çn,w) —>

(f«i,...,{„),«;), from Cnc(ô) x C^e) to Cnc+l(e).

In fact, if we choose, for a particular a = (ax, ... ,af), complex numbers

o)x , ... ,ojn such that

(5.10) a>j=oi for /' = 1, ... ,n,

and let

(5.11) ^„(y,, ... ,y„,z) = (©,>'■,... ,ea y ,0,

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REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS 431

1 .then \pa maps M into C^(6) x Cc(e), and

(5.12) <t>" = Goy/a.

We then have

(5.13)

p°(y, f) = p(tf(y, t)) = pC(G(ipa(y, t)))

m-\

-Pc{wa{y,t)) = tm + Y.pUy)tm"1=1

where

(5.14) p"=po<tf,a

Pi(yx,---,y„)=pf(rxykx,...,onykn)

pf(coxyx,...,conyn).

(5.15)

Alsom

(5.16) p°(y,t) = Y[(t-wai(y))i=i

where

(5.17) tü'öO = ^(WjP,, ... ,conyn).

Finally, if we let Aa(y) denote the discriminant of pa(y, •), it is clear that

Aa(y) = AC(œxyx, ... ,0)nyf)

(5.18) =Äc(coxyx, ... ,œnyn)œkayka

= Aa(y)yka,

where Aa(y) = Äc(coxyx , ... ,conyn)o'\

If we let K = O" (0), then Lemma 3.5 tells us that the germ of p at 0 can

be desingularized if the germ of p o <p at q can be desingularized for each q

in K, i.e. if the germ of p" at 0 can be desingularized for each o. If we

now drop the a , restrict outselves to the cube C^+1 (e), where £ = min(£, ô),

relabel m - I as m, ka as a and £ as £, and change the names of the y.

to x¡, we arrive at the following situation (remembering that one of our root

functions wt was the zero function):

(5.III.a) p: C^+ (e) ->R is a distinguished polynomial of the form

(5.19) tm+{+Pi{x)tm + ...+pjx)ti

where px, ... ,pm are real analytic functions on C^(e), that satisfy px(0) =■■=PJ0) = 0,

(5.HI.b) the discriminant A(x) of p(x,-) is of the form

(5.20) A(x) = A(x)xn

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432 H. J. SUSSMANN

for some a e Z" and some nowhere vanishing real-analytic function A:

C"(e)-H,

(5.III.C) there exist m complex-valued real-analytic functions wx, ... ,wm on

Cg(e), such that

m

(5.21) p(x,t) = tl[(t-wfx))1=1

for x e C"(e), t e]- e, e[.

We have shown so far that, if we prove that we can desingularize the germ at 0

of every p for which (5.III.a,b,c) above hold, then the general desingularization

theorem follows.

6. Properties of the global root case

We now begin the last part of our proof, in which we study a p for which

(5.III.a,b,c) hold, and prove that the germ at 0 of such a p can be desingularized.

The main point of the argument is that conditions (5.III.a,b,c) have strong

consequences about the root functions w¡, and imply in particular that the w¡

themselves must be desingularized. To see this, recall that, if P is a monic

polynomial of degree p, with roots Xx, ... ,X , then the discriminant A of P

satisfies

(6.1) A = l[(Xi-Xj).

If we apply this to our situation, with wx, ... ,wm being the functions of

(5.21), and w0 = 0, we see that all the differences wt - w., i = 0, ... ,m,

j = 0, ... ,m , i / j, divide A. In particular, the functions to,, ... ,wm

themselves divide A, because w.t = wi - wQ .

We will want to consider a slightly more general situation; namely:

(6. La) fis a real-analytic real-valued function on Cn+ (e),

(6.1.b) XeZ+ , peZ"+, aeZ\,

(6.1.c) g is a real-analytic real-valued function on C"(e),

(6.1.d) wx, ... ,w are real-analytic, complex-valued functions on C(e),

such that w¡ and w¡ - w. do not vanish identically on C(e) if i / j, i,je

{1,...,«},(6.Le) A: C"(e) -» R and B: C"+1(e) —» R are real-analytic nowhere vanish-

ing functions.

(6.I.Í) f(x,t) = B(x,t)fx'iY[f=x(t-wfx)) for x = (x,, ... ,xH) e C"(e),te]-e,e[,

(6.1.g) g(x) = A(x)xn for x e C(e), and

(6.1.h) the functions wx, ... ,wm and all the differences w¡ - w¡, i,j e

{I, ... ,m}, i ¿ j, divide g.

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REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS 433

We will prove

Lemma 6.1. Whenever (6.La,... , h) hold, then the germ off at 0 can be desin-

gularized.

This will in particular imply our conclusion because, if we take f = p , X= I,

p = 0, B = 1 , g = A, then conditions (6.La,... , h) are clearly satisfied, and

so the germ of p at 0 will turn out to be desingularizable.

To prove Lemma 6.1, we will use induction on \a\. If \a\ = 0, then the

functions w¡ must satisfy wfO) ^ 0. Hence, if we pick a small enough ô > 0,

the function

m

(6.2) B(x,t) = B(x,t)l\(t-wfx))i=l

never vanishes on Cn+ (a). Therefore

(6.3) f(x,t) = B(x,t)fxM on Cn+\â)

with B never vanishing. Hence / is desingularized.

Now suppose that

(6.II) k e Z+ is such that, whenever a system of data e,f,X,p,a,g,wx,...,

wm,A,B satisfy (6.La,... , h) and \a\ < k , then the germ of f at 0 is desin-

gularizable.

Assuming that (6.II) holds, we now pick a system of data e,f,X,p,

a,g,wx, ... , wm ,A,B for which (6.1.a,... , h) hold and |a| = k + 1, and

we prove that the germ of / at 0 can be desingularized. We begin with some

preliminary steps.

(6.Ill) If i e {I, ... ,m} and w: is not real-valued, then the complex conju-

gate function wT is one of the tu .

To see this, notice that (6.1.d) implies, in particular, that there is a nonempty

open set U ç C"(s) such that wfx) ^ wfx) for all j ^ I, x e U. Since /

and B are real valued, it follows that the polynomial function

m

t^H(t-Wj(x))y=i

is real valued for each x e U. This polynomial has m distinct roots wfx),

... ,wm(x). Pick an x° e U such that wfx0) <£ R. (If x° did not exist,

then w¡ would be real-valued on U, and hence on C"(fi), contradicting our

assumptions.) Then there is a jQ such that wfx0) = w. (x ), because the

nonreal roots of a real polynomial are conjugate in pairs. For x near x , there

is a j(x) such that wj{xfx) = wfx). By continuity, j(x) must equal jQ for

x near x , and so w. (x) = wfx) for x in some nonempty open subset of

C(e). By continuity, w. = W¡ on C"(e), and (6.Ill) is proved.

(6.IV) We may assume, in addition, that wfO) = 0 for i = I, ... ,m.

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434 H. J. SUSSMANN

Indeed, let I = {i:l < i < m, wfO) / 0}. Clearly, if / contains an i

such that u>( is not real valued, then it also contains a j such that w. = w¡.

Therefore the product

(6.4) B(x,t) = B(x,t)]\(t-wfx))iei

is real valued. By taking £ smaller, if necessary, we may assume that B(x, t) ^

0 for all (x,t)eC"+l(e). Then

(6.5) f{x , t) = B(x, t)fx" H(t- wfx)),i$i

which is exactly of the same form as the original expression for /, except that

the product only involves functions w¡ for which wfO) = 0.

Each w¡ is of the form wfx) = Dfx)xß{,) , where D¡:C"(e) —>

(6.V) C is a nowhere vanishing real-analytic function, and ß(i) =

(/?(/),, ... , ß(i)„) is a multi-index in Z" .

This is just a consequence of Lemma 3.4.

If y — (yx, • - ■ , y„), à = (<V • • • > <U are multi-indices in Z" , then we write

y <S if yx <ôx, ... ,yx <Sn.

(6 VI) For every i ,j , one of the inequalities ß(i) < ß(j) , ß(j) < ß(i)

holds.

To see this, we may assume that / / j. Since the difference w¡ - w¡ divides g,

then Wj — Wj = E(x)xy for some nowhere vanishing E and some y e Z" . The

Taylor series J2U rlx" °f w, at 0 is such that r' = 0 unless ß(i) < v . Also,

the series Y] rJx" of w. satisfies rJ = 0 unless v > ß(j). Then the series

YLp„xP °f w¡ - w, is given by pv = r'v - r'v , which shows that pv = 0 unless

v satisfies at least one of the inequalities v > ß(i), v > ß(j). But p -/ 0.

Hence y > ß(i) or y > ß(j). On the other hand, pv = 0 unless v > y. If

ß(i) 3t ß(j), then r!m = 0 and r'ß(i) ¿ 0, so ^(/) ^ 0. But then ;8(i) > y.

If /?(/) ^ /3(7), then it follows similarly that ß(j) > y. Since y > ß(i) or

7 > )ff(7"), we get, if ß(i) i ß(j), ß(i) £ ß(j), that either y > ß(i), in which

case ß(i) > y and ß(j) > y imply ß(j) > ß(i), or y > ß(j), in which case

ß(i) > y and ß(j) > y imply ß(i) > ß(j). In either case, we have reached

a contradiction, showing that ß(i) ^ ß(j) and ß(i) £ ß(j) cannot both hold.

This establishes (6.VI).

We now group the indices i into equivalence classes Ex, ... ,Er,by letting

two indices i ,j be equivalent if ß(i) = /?(/). (It may happen that r = 1.) We

let ß(Ef be the multi-index ß(i) that corresponds to all the / e E¡. After

relabelling the E¡, we may assume that

0<ß(Ex)<ß(E2)<-<ß(Er).

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REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS 435

Here "7 < <T means "7 < ô but "7 ^ <T. The inequality 0 < ß(Ex) follows

from the fact that all the w¡ vanish at 0.

Since ß(Ex) > 0, at least one of the components of the multi-index ß(Ex) is

t¿ 0. After relabelling the coordinates xx, ... ,xn,if necessary, we may assume

that the first component of ß(Ex ) is / 0. Hence, if we write

(6.7) ß(E,) = (y*(I), y(0), 7*(/) e Z+ , y(/) e Z+l,

then we have

(6.8) 0<7*O)<7*(2)<---<7>),

(6.9) 0 < y(l) < 7(2) <•••<?(/•),

and, for / = I, ... ,r - I, at least one of the inequalities y*(l) < y*(l + 1),

y (I) <y(l+I) is strict.

Write

(6.10.a) p = (px,p), px eZ+, peZf ,

(6. lO.b) a = (a, ,a), a{ eZ+, a eZ+ .

(6.10.C) x = (xx,x), x = (x2, ... ,xn).

Then f,g have the expressions

r

(6.11) fi(xx,x,t) = B(xx,x,t)fxfxß]\X[(t-Dfxx,x)xf(')xm),l=\ l€E,

(6.12) g(xx,x) = A(xx,x)x'x"xa,

for x e Cn~ (s), |x, I < £, \t\ < £ .

7. The blowups

We now want to blou up the origin of the (t ,x{) plane. We will not need

the general theory of blowups, which can be found in any algebraic geometry

textbook, but only the two-dimensional case. On the other hand, we have to

be precise about the local description of the blowups using coordinates, so we

begin by reviewing the standard definition and introducing some notations.

We use P, to denote one-dimensional real projective space, and L to denote

the canonical line bundle over P, , so P, is the set of all lines through the

origin in R and the fiber of L at a point L e Px is the line L itself. We

let P", P, denote, respectively, the set of lines other than the x (resp. y)

axis. If L e P°, then Ç(L) is the unique number x such that (x, 1) e L.

Similarly, if L e P, , then «(L) is the unique y such that (1 ,y) e L. Clearly,

(Pax,Ç) and (P, ,n) are coordinate patches covering P, , and ¿;,n are related

by £,(L) = Ifrj(L) on P" n P*. If 7t:L -> P[ is the projection, then we let

If = 7t"'(P,), if = Tr'(P^). If (p,L) e If, i.e. p e L eP\, then we

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436 H. J. SUSSMANN

let pa(p,L) be the y coordinate of p, and £,(p,L) = £(L). So (Ç,pa) is a

coordinate chart on if , that identifies if with R2. One defines /?A:L —► R2

similarly, and then the formulae relating the two charts on LanL are

(7.1) í«=l, Pa = Pbrl> Pb = PaZ-

We let F:L —> R be the map (p ,L) —> p . Then F is a proper Cw map

which is a diffeomorphism from the complement of the zero section of L onto

R - {0}, and maps the zero section—which is diffeomorphic to P, , i.e. to a

circle—to the origin of R .In coordinates, F is given by

(7.2) x = PaZ, y = Pa onLfl,

(7.3) y = Phr], x = Pb onL6.

Now let

(7.4) L(£) = F-'(C2(£)), F£ = F[L(£).

Then F£ is proper, and F£"'(0) = F~l(0). We let L°(e) = L(e) n La and

L (fi) = L(fi) nL . Then lf(e) is the subset of if characterized, in terms of

the coordinates (£, pa), by the inequalities

(7-5) \Pa\<e, \Pai\<e.

Similarly, L2(e) is characterized by \pb\ < e , \pbr¡\ < e .

We now identify Cn+[(e) with C2(e) x C"~fe) by means of the map

(x,t)^((t,xx),x).

We then define

d>:L(£)xC""'(e)^C"+'(£)

by

(7.6) ®((p,L),x) = (Ffp,L),x).

In coordinates, the map <P is given by

(7.7.a) (Ç,pa,x2, ... ,xn)-*(pat,pa,x2,... ,xn)

on La(fi) x C"~ (s), and

(7.7.b) (rj,pb,x2, ... ,xn)^(pb,pbr\,x2, ... ,xj

on L2(e) x Cn~ (e).

Let M = L(e) x C"~l(e). We have defined a C'° map from M onto

C"+'(fi), which is proper and surjective. If we let K = F~ (0) x {0} , then

K = <P~'(0). In view of Lemma 2.6, it will follow that the germ of / at 0 is

desingularizable, if we prove that, for each point p of K , the germ of / o <p

at p is desingularizable.

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REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS 437

Let / = fo <p. We write the expression of / on L2(e) x Cn~ (e), in terms

of the coordinates £,pa,x2, ... ,xn. Let B = B o <p. Then we get

(7.8) f(i,pa,x) = B(^,pa,x)p^exßf[]J(p^-Dfpa,x)pl'{l)xnn).<=1 !'€£■/

Using this, we prove that, if p is a point in K, then the germ of / at p is

desingularizable. We treat separately the cases:

(A) peV(e)xCn-fe),(B) peLb(e)xCn~l(e).

We treat case (A) first. Let the Ç, pa coordinates of p equal cf , p°, respec-

tively. Then p° must be equal to zero. Since all the y*(l) are strictly greater

than zero, we may factor pa from each factor in the product that appears in

(7.8), and get (with y (I) = y*(l) - 1) :

(7.9) m , Pa, x)=a«; , Pa,x)Pxa+m+^xß n n (í - D^a. ^^f.l=\ i€E,

Let us distinguish the subcases:

(Al) 7(1) >0 or 7(1) >0,

(A2) 7(1) = 0 and 7(1) = 0.

We consider (Al) first. In this subcase, we distinguish two sub-subcases,

namely:

(Ala) ¿Vi),(Alb) {° = 0.

In sub-subcase (Ala), the function

(7.10) H(t:,pa,x) = B(i,pa,x)c;lf[]J(i-Dfpa,x)pl{l)xñl))1=1 i€E,

has the value

(7.11) /í(¿:0,o,0)(c:0r+A

at (£ ,0,0), i.e. H does not vanish at p. Therefore, in a neighborhood of

p, f is equal to the product of the nowhere vanishing function H, and the

monomial p'f+/-+fl,x^ . Therefore the germ of / at p is desingularized.

We next consider sub-subcase (Alb), i.e. we assume that Ç = 0. Pick

¡€{l,...,m}, and let / e E¡. Let

(7.12) w*(pa,x) = Dfpa,x)pfxm.

Then wfpa ,x) = pqwj (pa,x). Let us define

(7.13) g#(pa,x) = A(pa,x)pf-{x\

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438 H. J. SUSSMANN

(Notice that a, > 1 , because the w¡ divide g and therefore y*(l) < a,.)

Then, in a neighborhood of (0,0,0), /(£,pa,x) has the expression

(7.14) f(c:,pa,x) = B(t;,pa,x)pXa+m+ßiclXxiiY[Y[tt-w*(pa,x)),

where the functions wi vanish at 0, and the wf and wj - w. divide g .

That is, / satisfies the same conditions as /, except that (ax, ... ,an) is

replaced by (a, - 1 ,a2, ... ,an). By the inductive hypothesis, the germ of

/ is desingularizable at 0. This complete the consideration of sub-subcases

(Ala,b), and therefore of subcase (Al).

We now consider the remaining subcase of case (A), namely subcase (A2).

That is, we now assume that y(l) = 0, y (I) = 0, i.e. that the index ß(Ex) is

just (1,0, ... ,0). In this subcase, the expression for / on lf(e) x C"~ (e)

becomes (in terms of the coordinates £,pa,x2, ... ,xn) :

fi(í,Pa,x) = B(í,pa,x)pa 7r H(tA:-Dfpa,x))i'€£|

)■(/)-?(')>L 1=2 i€E,

We distinguish three sub-subcases, namely

(A2a) ¿r° = 0,(A2b) (f ¿ 0 and cf ¿ Df0,0) for all i € Ex,

(A2c) 0 / {° = DfO, 0) for some / e Ex .Sub-subcase (A2a) is essentially identical to sub-subcase (Alb). In a neighbor-

hood of (0,0,0), the product Yi,eE (£~Di(Pa >•*)) ̂ oes not vanish. Therefore,

if we let B* equal B times this product, we have

(7.15) /(^/».,J0 = Ä^^/^.,^m+V^^^«-öl(/'.^/?0ÄÄI,)•1=2 i€E,

Now y(2) > 0 or y (2) > 0 . If we let w¡ (pa,x) be defined, for / e E¡, I >

2, by Formula (7.12), then the functions wj vanish at 0, and the w¡ ,wt -Wj

divide g (where g is defined by (7.13)). Moreover:

(7.16) f(i,paJ) = B*(^pa,x)pl+^^xßf[l[l^-w*(pa,x)).1=2 i€E,

By the inductive hypothesis, the germ of / at (0,0,0) is desingularizable.

Next we turn to sub-subcase (A2b). In this case, if we let

r ~

(7.17) B(^,pa,x) = B(i,pa,x)èl[ll(i-Dfpa,x)pfl)xm),l=\ i€E,

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REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS 439

then B does not vanish in a neighborhood of (¿j ,0,0). Therefore

(7.18) M,pa,x) = B(c-,pa,x)p"a+m+ß'xß

near (¿;0,0,0), with B not vanishing. Then the germ of / at p is desingu-

larized.

Next we dispose of sub-subcase (A2c). We let F denote the subset of Ex

that consists of those indices i for which Df0,0)=£, . If we let

(7.19) B(t; ,pa,x) = È(t,pa,xKA fi (í - D¿Pa >^P'T*

where E* = (F, -F°)uF2U- • uFf, then B does not vanish in a neighborhood

of (Ç0,0,0). The function / has, near (¿;0,0,0), the expression

(7.20) f(C,pa,x) = B(C,pa,x)pXa+m+fl<xßY\(cA:-Dfpa,x)).

iGE°

Assume first that one of the functions D¡, i e E , is real valued. Pick an i

for which this happens, and call it iQ . The map

(7.21) (i-,pa,x)^(cl-Dio(pa,x),pa,x)

is nonsingular at (£°,0,0), and maps (¡f0,0,0) to (0,0,0). If we use 9 for

the new coordinate £ - Di (pa, x), then / can be expressed, in terms of the

(9,pa,x) coordinates as

f(9,pa,x) = B(9,pa,x)p*a+m+f"xß9 T] (9-Dfpa,x))

i€E"

it?

where, for i e E , i ^ i°, we define

(7.22) Dfpa,x) = Dfpa,5c)-Dlo(pa,x).

If i e E , the function PaDfpa,x) is equal to w¡, because we are assuming

that ß(Ex) = (1,0, ... ,0). Hence paDx is equal to w: - wi . So, if i ^ z'0,

and we define g as in (7.13), we find that paDfpa,x) divides g(pa,x), and

therefore Di divides g . Similarly, paD¡ - paDj equals w¡ - w¡. So, if i ^ j ,

i e E , j e E , then paDi - paD. divides g, and therefore Dj - D- divides

g#. Finally, since all the Df0,0) have the value ¿;0 for i e E° , it is clear that

the D¡ vanish at 0. This show that, in a neighborhood of (Ç , 0,0), / satisfies

the same conditions as /, but with a replaced by (ax-l ,a2, ... ,an). Then,

by the inductive hypothesis, the germ of / at (£ ,0,0) can be desingularized.

We now assume that none of the functions D¡, i e E , is real valued. Pick

an ir.eE . Then w. (p„,x) = p D. (pa,x). We know that the conjugate w~\J IQ Li Li IQ LI IQ

is one of the functions w . Let W = w . Then w. = p D (p ,x), where( In t\ I] ' a /| v' d '

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440 H. J. SUSSMANN

Djt = D¡o. Clearly, /, is also in F, , and D¡ (0,0) = £° , showing that ix e E° .

Since no difference Wj-w., / / j, vanishes identically, the index /', is unique.

Since the D¡, i e E , are not real, we have i, / /'0 .

Therefore we can partition E into a disjoint union of pairs {/,7'} , i ^ j,

such that D = D¡. Form a set EQ by picking one index i from each such

pair. We can then rewrite Formula (7.20) as follows:

(7.23) M ,Pa,x) = m, pa , x)pXffm^ xß H [(Í - Dfpa , x))(C - Dfpa,x))}.ieE°

Since all the functions w¡, w¡ - w¡ divide g, it follows that, for i ,j e EQ ,

the functions D¡ - D., Z); - D divide g# . For each i e F0 , write

(7.24) Dj(pa,x) = afpa,x) + ^l~bj(pa,x),

where the a¡, b¡ are real-valued functions. Then the a¡, bt satisfy

(7.25) û|.(0,0)=^V0, ¿,(0,0) = 0.

For each i e E0 , the function D¡ - D¡ divides g*, and therefore it is,

by Lemma 3.1, a product of a monomial times a nowhere vanishing function.

Therefore we can write (using y for (pa ,x)) :

(7.26) bfy) = bfy)ym

for some £(/) e Z" , and some function bt which is Cw, real valued and

nowhere vanishing near 0.

Of all the multi-indices £(/) defined in this fashion, choose a £(/0) which is

maximal, i.e. such that, for every i e EQ , it is not the case that Ç(i) > í('0).

We claim that

,-, n the functions D, - a, , D, - a, , i e En , divide g , in a( 1.1) ' '0 ' '0 <J

neighborhood of (0,0).

To prove this, observe first that the conclusion is trivial if i = i0, for in this

case D -a is just JAA\b , which we know divides g . Now let ij=in. The' '0 '0 ' u

functions D-D , D -D divide g , and so does their difference D.-D, .

Therefore, by a reasoning identical to the one used to prove (6.VI), we can write

(7.27) D¡-Dio = Qiy"(l),

(7-28) Dk - Dk = Ry"

for some nonvanishing Q¡, R and multi-indices «(/), o that satisfy, moreover,

the condition that either «(/) > ct or «(/) < a . The function R must then be

-2\fAA\b. , and o has to be C(/0). Now

D.-fl. =D.-Di +v/rTè,

( } = Q,y"in + v^T bt/'°\

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REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS 441

Since Ç(i0) = a, one of the following occurs: (1) Ç(iQ) > «(/'), (2) Ç(i0) <

n(i), (3) £(/„) = i/(/). If (1) holds then

(7.30) Dl-ah = [Ql + 0{\y\)ly*i).

If (2) holds, then

(7.31) Di-ak = [y/=îbh + 0{y\]ym.

So, in either case, D¡ - a¡ is equal to a nonvanishing function times a mono-

mial that divides g# . (That the monomials y"1^ , y ('o) divide g* follows from

(7.27), (7.28), C(/0) = o , and the fact that D¡ - Dio, D¡o - ZX divide g*.)If (3) holds, then

(7.32) D¡-a,={Qt + ,/=íb.)yC(k)

If we write Ql = qt + v^T^, , with q¡ , q¡ real, and equate imaginary parts in

(7.32), we find

(7.33) bi = (q2i+iy(k).

If we compare this with the expression

(7.34) b, = b,ym

and use bfO) ^ 0, we conclude that Ç(/0) < Ç(i). By the maximality of Ç(i0),

we then have Ç(z) = Ç(i0). But then b¡ = q¡ + b¡o. In particular, this implies

„ido)

2

hthat (q. + bj)(0) ¿ 0. We then rewrite (7.32) as

(7.35) Di-aio = QiyC

where Qx = q) + y/-l(qf + bio). Since (qf+ b¡0)(0) ¿ 0, it follows that ß(.(0) ̂

0. But then D -a divides g .I ¡Q °

This completes the proof that the £>/ —a/ divide ^ . The proof that Di-al

divides g is exactly the same.

Having proved (7.1), we now state the much easier fact that

(7.II) any difference oj'two distinct functions D(. - a; , Dx. — a¡ , ie

F0 , divides g near 0.

This is so because any such difference is of the form D¡ - D or Dt - D ,

and we already know that these differences divide g .

We now let

(7.36) üt = Dt-*k

for all i e E . (It is now convenient to return to the labelling by F , rather

than F0 .) Also, we make the change of coordinates

(£,pa,x)^(£\-ai(pa,x),pa,x),

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442 H. J. SUSSMANN

which is nonsingular, and use 9 for the new coordinate Ç-ai(pa,x) and f ,B

for the functions /, B expressed in terms of 9 , pa , x. Then the expression

of / in the new coordinates is

(7.37) j(9,pa,Ai) = B(9,pa,x)p^m^xß^\(9-wfpa,x)).i€£°

Since the wi and their differences wt - w. divide g , we can apply the

inductive hypothesis and conclude, once again, that the germ of /o<p at p can

be desingularized.

This concludes the discussion of all the subcases of case (A). We now consider

case (B). In this case, the expression / of /o$ in the coordinates (r\,pb,x)

is

kn>pb,x) = £(",/v-*)/9¿+/"'/'''-*''

(7.38) xnn^-^M.v;v,(,)^)l=\ i€E,

i.e.

f(fl,Pb,x) = B(r],pb,x)pb n x

(7.39) ,)-(/) „7* (/)-?(/)>,*YIY[{\-Dfpb«,x)pl()ny()*y

1=1 ¡€£,

The point p has coordinates (« ,pb,0), and pb clearly equals 0. If « / 0,

then p e L2(fi) x Cn~(e) as well, and we already know from case (A) that the

germ of / o <I> at p can be desingularized. So we need only worry about the

case when r¡° = 0. But, in that case when the product that appears in (7.39)

cannot vanish near p, due to the fact that y*(l) > 0. Hence /, near p, is

the product of a nonvanshing function times /?¿+//2+/'«/ilx/', and therefore the

germ of / o <p at p is desingularized.

This completes the consideration of the last remaining case. The proof of

the weak desingularization theorem is now complete, d

8. Subanalytic sets

We now describe how the weak desingularization theorem can be used to

prove the basic facts about subanalytic sets.

Let M be a real analytic manifold, and let S be a subset of M. We say

that S is semianalytic in M (and write S e SMAN(M)) if every p e M

has a neighborhood U such that there exists a finite set AT of real-valued Cw

functions on U with the property that S n U belongs to the algebra 3§(F)

of subsets of U generated by the sets {x:f(x) = 0}, {x:f(x) > 0}, f e

AT. A subanalytic subset of M is a subset 5 such that there exists a triple

(N,T,tp) that satisfies: (i) A is a Cw manifold, (ii) T eSMAN(N), (iii)

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REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS 443

<t> e CW(N ,M), (iv) <j> is proper on the closure of T, and (v) 0(F) = S. We

use SBAN(M) to denote the class of all subanalytic subsets of M.

Let S e SMAN(M). For each p e M we can choose a neighborhood U

of p and a finite subset AT(p) of CCÜ(c//, ,R) such that SnUpe A%(AT(p)).

We can then choose a toric desingularization (N , 9 , v ) of AT(p), such that

v has full rank. If Q is a connected component of Np , identify Q with T"

via 9 . Let 9>x be the partition of Sl into the four sets {9:0 < 9 < n},

{9:n<9 < 2n}, {0} and {n}. Let ATn be the partition of T" into 4" sets,

whose elements are the products /, x • • • x In , Ij e ATX. Let ATn(Q) be the

partition of Q obtained from ATn via 6 . If Z e ATn(Q), then every sine

monomial on (A, 9 ) either vanishes identically or nowhere on Z. Hence

all the sets i> (Z), Z e ATn(Q), Q e <ff(Np), are either entirely contained in

or disjoint from every set {x:x e U ,f(x) = 0}, {x:x e U ,f(x) > 0},

for every f e AT(p). Therefore every set in AM(AT(p)) is the union of those

sets v (Z) that intersect it. In particular, this is true for S. By an obvious

paracompactness argument, it follows that S is a locally finite union of sets of

the form v(Z), for some Z e ATn , and some v e Cw(Tn ,M). Equivalently,

S is a locally finite union of cubic images, where a cubic image is a set of the

form h(Cm(l)), for some m e Z+ and some h e Cw(Cm( 1 +e),M), for some

e > 0. Moreover, if we call a cubic image h(Cm(l)) an m-cubic image, we see

that 5 is a locally finite union of sets S¡ that are m-cubic images for integers

m such that m < dim M.

Since all the maps v can be taken to be of full rank, we see that the m

can be taken to be strictly less than dim M, if S has empty interior. Since an

m-cubic image is an m'-cubic image for any m > m , we have shown that

every S e SMAN(M) is a locally finite union of n-cubic images

(a) (where « = dim M) , and of (n -1 )-cubic images if S has empty

interior.

Since subanalytic sets are, by definition, proper images of subanalytic sets, a

conclusion similar to (a) is also true for them, except that, for S e SBAN(M),

S = f(T), Te SMAN(N), f e CW(N,M), f proper on the closure of T,

the number « might have to be replaced by dim T which may, in principle, be

much higher than dim M. One can easily show, however, that is not the case.

In fact, if we define the dimension of a subset A of a Cw manifold M to be

the largest of the dimensions of all connected Cw submanifolds B of M such

that B C A , then we have

,, , if S e SBAN(M) , and dim S = k , then S is a locally finiteunion of k-cubic images.

To see this, it suffices to prove that, if fie Cw(Cm(l + e),M) and df has

rank < m everywhere, then /(Cm(l)) is a finite union of (m- l)-cubic images.

This will follow from (a), if we find a semianalytic subset A of Cm(l +£) such

that «(vinCm(l)) = «(Cw(l)),and A has empty interior in Cm(l+£). To find

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444 H. J. SUSSMANN

A , let p = max{rankdf(p):p e Cm(l + s)} , and pick ape Cm(l) where df

has rank p. Construct a Cw function 4>:R'n —> R which vanishes on 9Cm(l),

is strictly positive on Cm(l), and is such that F = (f ,tp) has rank p + 1 at

P.We let A = {x:x e Cm(l), rankdF(x) < p+ 1}. Then A is semianalytic in

Cm(l + e) and has empty interior. If x e Cm(l), then either rankdf(x) < p ,

in which case x e A , or rankdf(x) = p, in which case we can let L(x) denote

the leaf through x of the foliation induced by the submersion / on the open

set n = {y: yeCm(l), rank df(y) = p}.

If a = sup{(p(y): y e L(x)}, and {x } is a sequence of points of L(x) such

that </>(Xj) —► a, we can pass to a subsequence and assume that x —> x ,

x e ClosC'"(l). Then a = (p(x). Since a > <f>(x) > 0, it follows that

x e Cm(l). If x e L(x), then x is a maximum of the restriction of </> to

L(x), so that dcp(x') is a linear combination of the dffx), where the f. are

the components of /. But then rankdF(x') = p, and x e A . If x £ L(x),

then x e A. So, in all cases, there is an x e A such that h(x') = h(x).

Therefore h(A) = h(Cm(I)). So (b) is proved.

In particular, (b) implies that every locally finite union of subanalytic subsets

of M is subanalytic. If S¡ = ffTf , T¡ e SMAN(N¡), i =1,2, then Sx nS2 =g(Z), where Z c A, x A2 is given by Z = (Tx x Tf) n{(x,y):fx(x) = f2(y)},

and g(x,y) = fx(x). So SBAN(M) is also closed under finite intersections.

It also follows from (b) that a one-dimensional S e SBAN(M) is a locally

fintie union of sets Ah = {h(t):0 < t < 1}, h e Cw((-e,l +e),M). In

coordinates, a set Ah is given by x; = h ft), i = 1,...,«, where the h¡ are

analytic on (-e, 1 + e). From this, it follows easily that Ah is semianalytic.

Therefore

(c) every one-dimensional subanalytic set is semianalytic.

Moreover, (b) also implies

, j, if A e SBAN(M) , then the set of connected components of A

is locally finite, and each component of A is subanalytic.

If A = h(Ck(l)), «eCw(C*(l+£),M),thentheset Q of those peCk(l)

where dh has rank k is open in R . Clearly, Q is then a finite or countable

union of open sets Q, such that each set h(Clf is a connected embedded

Cw ^-dimensional submanifold of M. So h(Cl) is a finite or countable union

of connected C"' submanifolds of M. Since A = A(Q) U Ax , where Ax =

A(C*(1) - Q), so that Ax e SBAN(M) and dimAx < k , an easy induction

shows that A is a finite or countable union of connected Cw submanifolds of

M. Once this is known for cubic images, it follows that

if A e SBAN(M) , then A is a finite or countable union of

connected C(" submanifolds of M.

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REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS 445

(Naturally, the stratification theorems proved in the next section will imply

the much stronger conclusion that the union can be taken to be a locally finite

partition.)

9. Stratifications and complements

We are now ready to prove the basic theorems on the existence of stratifi-

cations compatible with subanalytic sets and maps and having various special

properties. As a by-product, we will also obtain the theorem that the comple-

ment of a subanalytic set is subanalytic.

Let M be a Cw manifold. A stratum in M is a connected, embedded,

Cw submanifold of M. A stratification in M is a set AT of pairwise disjoint

strata in M which is locally finite (i.e. every compact subset of M meets

at most finitely many members of AT), and satisfies the frontier property, i.e.

the property that, whenever S, T are members of AT, such that T ^ S but

T n Clos 5" t¿ 0, then T ç Clos S and dim T < dim S. Clearly, every subset

of a stratification is a stratification.

If ¿rf is any family of sets, we write [sf\ to denote the set \jssi . If AT

is a stratification in M, then the support of AT is the set \AT\. If k is a

nonnegative integer, we use AT to denote the set of all ¿c-dimensional strata

of AT. Then \AT | is a k-dimensional embedded Cw submanifold of M,

whose connected components are the members of 5" .

If A, S c M, we call S compatible with A if either S c A or SnA = 0. If

AT, sT are collections of sets, we call AT compatible with sé if every S e AT

is compatible with every A esf . If AT is compatible with a set A , then we

use AT\A to denote the set {S: S e &, S ç A} .

If AT is a stratification and A = \AT\, then we say that AT is a stratification

of A . Clearly, if A c \AT\ and AT is compatible with A , it follows that AT\Ais a stratification of A .

A block in M is a relatively compact stratum S in M such that there exists

a Cm surjective diffeomorphism <P: C —* S—where C is the open unit cube in

R , and k = dim S—such that the graph G(Q>) of <P is subanalytic in R x M.

Since S = 7t(G(Q>)), where n is the projection from R x M to M, it follows

that S is subanalytic in M. If M, A are Cw manifolds, f eCw(M,N),

S ç M, T ç A, and F is a block in A, we say that S is a block lift of T

by f if S is a stratum in M, as well as a relatively compact subset of M,

and there is a Cw surjective diffeomorphism 4>: C x T —> S, where C is the

unit open cube in R" , and v = dim S - dim F, such that f($>(c,t)) = t for

(c, t) e C x T, and C7(0>) e SBAN((R" x N) x M). (This implies in particular

that f(S) = T, and that 5 is a block in M.) if AT, AT are stratifications in

M,N, respectively, such that AT consists of blocks, we say that AT is a block

lift of F by / if every S e y is a block lift by / of some F e y .

If 51 is a block in M, and p e M, we say that S is locally connected at p

if every neighborhood U of p in M contains a neighborhood V such that

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446 H. J. SUSSMANN

VnS is connected. If S ç A ç M, we call S A-locally connected if it is locally

connected at every point of A . (Notice that the local connectedness condition

is trivially satisfied if p e S or p £ Clos S, so one only needs to verify local

connectedness for p e A n (Clos S - S).) If A = M, we simply call S locally

connected.

The stratification results are

Theorem 9.1. Let M be a Cw manifold, and let se ç SBAN(M) be locally

finite. Then there exists a stratification AT of M by locally connected blocks

which is compatible with sé .

Theorem 9.2. Let M,N be Cw manifolds, and let sé c SBAN(M), 3§ ç

SBAN(N) be locally finite. Let f e CW(M, A), and let L e SBAN(M) beclosed and such that fis proper on L. Then there exist stratifications AT, AT of

M, N, respectively, such that AT is compatible with séu{L}, AT is compatible

with ATA, AT consists of locally connected blocks, AT consists of blocks, and AT \L

is a block lift of AT by f.

Clearly, Theorem 9.1 is a particular case of Theorem 9.2 but it will be con-

venient to prove it separately. In particular, Theorem 9.1 can be applied to

sé = {A}, where A e SBAN(M). It then follows that M-A is a locally finite

union of blocks, and in particular a subanalytic set. So Theorem 9.1 implies in

particular:

Corollary 9.3. If M is a Cw manifold M and A e SBAN(M), then M -Ae

SBAN(M).

To prove the stratification results, it is convenient to introduce some prelim-

inary definitions. Suppose that k , C, Q, 4>, M are such that C is an open

cube in R , Q is a neighborhood of Clos C in R , M is a Cw manifold,

and 0 is a Cw map from Q to M such that dcf>(p) has rank k at some point

of C. Let A = cp(C). Then we call A a k-dimensional cubic image in M,

and we refer to the triple (C,£i,tp) as a realization of A. If (C,Q,4>) is a

realization of A , we use B(C, tp) to denote the union of Clos C- C and the set

of those p e C where dtp has rank < k . It is clear that B(C ,tp) is a compact

semianalytic subset of ÍÍ, so the set A = <fi(B(C,</>)) is compact subanalytic

in M. Clearly, dim ,4 < dim A, and A ç Clos .4. If p e C, p $ B(C,cf>),

then 4> has rank k at p, and so 4> maps a neighborhood of p in C to a

^-dimensional embedded Cw submanifold of M. We let JA^(p) be the germ

at (p(p) of this submanifold.

Suppose that, for i = 1,2, (C¡,Q¡,<fr¡) are realizations of /c(-dimensional

cubic images A,. Let k > ma\(kx ,kf). We define CrK(Cx,4>x,C2,<ft2) tobe

the set of those pairs (q,r)e (Cx x C2) such that rankd<f>f(q) = rankd(f>2(r) =

K, <f)x(q) = (f)2(r), but JA fq} ± J[ (f).

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real-analytic desingularization and subanalytic sets 447

Lemma 9.4. Let M be a Cw manifold and, for i= 1,2, let (C(. ,£2,. ,^(.) be

realizations of k¡-dimensional cubic images A; in M. Let k > max(rCj ,kf).

Then:

(A) CrK(Cx ,<p'x,C2,4>2) is a semianalytic subset of Clx x Cl2, of dimension

(B) CrK(C, ,4>x ,C2,(f>2) is closed in

Q, xil2-((B(Cx,4>x)x Clos C2)U (Clos Cx x B(C2,<p2))).

Proof. First we prove (B). Suppose pj = (q^rf) e CrK(Cx,(ßx,C2,(p2),

limJ^oopj = p = (q,r) e Q, x Q2 - ((B(Cx,cf>x) x ClosC,) U (ClosC, x

B(C2,<p2))). Then q $ B(CX,4>X) and r £ B(C2 ,<t>2), so q e C, , reC2,

and rankdcjyfq) = rankdtf>2(q) = k. If JA^fq) = Jx (r), then it is clear

that there is a diffeomorphism 9 from a neighborhood Vx of q in C, onto

a neighborhood U2 of r in C2, such that <p2(9(q')) = 4>x(q') for q e Ux .

Moreover, we can assume that 0( is one-to-one on U¿ for i = 1,2. Since

(o , rA) e CrK(Cx ,(j>x,C2, (¡)2), we have in particular <px (qf) = <p2(rf. But then

r; = 9(qf if j is large enough. Therefore J, (qf = J, (rf, contradicting

the fact that (q}, rf e CrK(C,, (¡>x, C2, <p2).

We now prove the semianalyticity of CrK(Cx ,(¡>{, C2, <p2). It is clear that we

can write each C( as a finite union of open cubes C , with the property that,

for every s,s , either <px(CXs)(~\<p'2(C2s,) = 0 or 4>x(CXs) U(j)2(C2s,) is a subset

of the domain of a cubic coordinate chart of M. From this it follows easily

that we may assume that M = R." . Also, we may assume that k = kx = k2,

for otherwise the set Cr^C, , </>, , C2,02) is empty. Let D ç C x C be the set

of those pairs (¿;, ,<J;2) suchthat (¡>x(¿,x) = (¡>2(£,2) and dtpfZf has rank* for

/ = 1,2. If we let A, denote the sum of the squares of the k x k minors of

the Jacobian matrix of 0(, then D is the subset of C x C characterized by

4>x (if - 02(^2)' ^ ,(<!;,) ̂ 0> A2(£2) ̂ 0, so D is a semianalytic relatively

compact subset of Ùx x Q7.

If 5 is a subset of {1, ... ,«} such that card(S) = k , we let S' = {I, ... ,

n}-S and, if S = {ix, ... , ik}, S' = {jx, ... , jn_k} , /, < i2 < ■ ■ ■ < ik , jx <

j2 < ■ ■ ■ < Jn-k ' then we let ns(xx, ... ,xn) = (xh,... ,x¡f), ns,(xx, ... ,xf) =

(x , ... ,x, _ ), so that x —► (ns(x) ,ns,(x)) is an isomorphism between R"

and R x R"~ . Let </>( s = ns o cp.. If d4>f£,) has rank«, then the germ

JA..(() is transversal to {(ns o (^^(c;)} x R"~ iff d<j>ls(i) has rank«:. So, if

(Çx ,Ç2) e D, and Jlt>fÇx) = J^fíf), then d<px S(Z,X) has rank k for a subset

S iff d(j>2 5(£2) does. Define Ds to be the set of those (£, ,£2) e D such that

both d<f>x S(ÇX) and d<j>2 s(£,2) have rank*. If A( 5 denotes the determinant

of the Jacobian matrix of 0( 5 , then Ds is the subset of C x C defined by

0i(£1) = 02(^2)' Ai s^f ^ 0> ^25(^2) ^ ^. Therefore, Ds is semianalytic

and relatively compact in Q.x x Q2 .

It is clear that CrK(Cx,<j>x;C2,^2) ç D. On the other hand, if (£,, ¿;2) € Z>,

there is always an S such that A, S(ÇX) ± 0. If A2 s(£,f) = 0, then J, (Çx)

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448 H. J. SUSSMANN

is transversal to {</>, S(ÇX)} x R"~ but JA, (£2) is not. Therefore JA. (£,) ^

JA^fÇf), and so (£,,£2) s CrK(C, ,0, ;C2,02). Hence the semianalytic set

Z) - (Us^s) is entirely contained in CrK(Cx ,<f>x ;C2,<f>2). So, to prove that

Cr^C, ,4>X;C2,4>2) is semianalytic, it suffices to prove that each Es is semi-

analytic, where

(9.1) Es = CrK{Cí,<pí;C1,<p2)nDs.

Clearly, it suffices to consider the case when S = {I, ... ,k}. Write 0 •(<!;) =

(\pfÇ), 9fi;)), where y/. = nso cf>., 9t = ns, o <f>.. Let n = (£, ,£,f) e Ds . Then

0i(íi) = 02(£2)- Let x = <t>fix). Then x = (x',x"), x e Rk, x" e Rn~k .

Both JAq (Çx), J,(£2) are germs of k-dimensional Cw manifolds through

x, transversal to {x'} x R"~ . Hence, for / = 1,2, JA^fÇf is given, in a

neighborhood of x , as the graph of a Cw function «(., defined near x , with

values in R"~ , and such that hfx) = x". The functions hx,h2 depend

on n, and we will emphasize this by writing hf,hn2. The germs JA, (¡A,x),

JA^fÇf) coincide iff hnx and h2 coincide on a neighborhood of x . Since

h\(x) = h\(x), we have J^f = J^fZf if and only if Dhf and Dh\agree on a neighborhood of x . (Here "D" stands for "Jacobian matrix of.)

The functions hA] satisfy

(9.2) df^) = hni(ipflA))

for £ near Iet. This implies that

(9.3) DefÇ)=Dh1(ipft;))-Dy,fcl)

for £, near ¡A¡. Therefore

(9.4) Dh1(y,fcl)) = DdfZ) ■ [Dipfd)]-X,

so that, if h) = (ti\ , , ... , h"ik), we have, for j = I , ... ,k , I = 1, ... , k ,

dh,i »///(Í)

where PT . ¡ is a Cw real-valued function on Q(. We claim that, if a =

(ax , ... ,ak) is a Ac-tuple of nonnegative integers such that \a\ = a, H-\-ak >

1 , then there exists analytic functions Z on Í2, such thati ,J ,a i

Ö1"1«" . Z. . ({)

(9-6) a^-aV(y'tf)) = Z^Kt

for ^ near £(.. (The functions Z; Q do not depend on «.) Indeed, we

have already established this for \a\ = 1 . If |q| > 1 , and the Z( „ have

been defined for \ß\ = \a\ - 1 , pick / such that a¡ > 0, and let ß =

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real-analytic desingularization and subanalytic sets 449

(a,, ... ,a,_x,al+x,... ,ak). Then, if we write £ = (Çx,... ,Çk), y/fO

(yiA(Ç), ... ,Vitk({)) we have, for m = 1, ... ,k:

where

8Zt . g 3A,, ç(9-8) WiJifjm = -^ ■ Ai<s - (2\ß\ - l)ZIJtß-

'm

So

or -i,5 v i/-1 ' i,j,ß oç^i,m 'i?

/ ô"V. \ dip. W . „ (<%)u I l J I /. . tz\\ Ti ,r r r\ i ,i ,p ,m^>

' a / d'H'h' . \ dtp. w. . „

<9'9) ^MwAM>ß))-k{i)--tM'If (ir„,(0) is the matrix of the complementary minors of (dy/j r(Ç)/dÇm)r m ,

so that

m=\ ^m

then we have

So, if we let r = /, we get the desired expression, with

(9-11) Zu,« = ibWij,ß,mtmrm=\

In particular, we have W[ = h2 in a neighborhood of x if and only if the

equalities

fO|21 Zl,j,cf^\) Z2,j,J^l)[- } A^«,)2"-"- A2^2)2H-'

hold for 7 = 1,..., re and all multi-indices a . For each j', a , we can define a

C™ function X. a:Qx x Q2 —» R by letting

(9.13) A^«,,«2)=^^,,|",Z/>/i(,«1)-AIii«1)2W-IZlt/ia(<2).

Then Fs is exactly the set of those « e Q, x Q2 such that (a) r¡ eDs, and

(b) A7>Q(>?) = 0 for all j,a. So

(9.14) F5 = Z)5n(nz(^,j)-J .a

The set P]; 0Z(i J is an analytic subset of Q, x Q2. So Es is semianalytic.

As explained above, this shows that Cr^C, ,4>X;C2,4>2) is semianalyitc in Qx x

a,.

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450 H. J. SUSSMANN

We now show that

(9.15) dimCrK(Cx,<j>x;C2,<¡>2)<K.

Assume that this is not true. Then there exists a nonempty K-dimensional

connected embedded Cw submanifold // of fi, x Í22 which is contained

inCrK(Cx ,4>x;C2,<f>2). In particular, this implies that <px(Çx) = 02(£2) when-

ever (£, ,Ç2) e H. Pick a point Jj = (£, ,£2) e H, and let v = (vx ,v2) be a

tangent vector to H at JJ. If / —► (Çx(t),Ç2(t)) is a curve which goes through

Jj when / = 0, and is contained in H, and such that <*.(0) = vj for i = 1,2,

we have 0,(^(0) = 02(Í2(O)> and so d^flf^fO)) = d<t>2(l2)(U0)).Since d(pfeif are injective for i = 1,2, we conclude that vx =0 iff v2 = 0.

Therefore, if we let n' denote the projection (Çx ,£,f) —► Ç., restricted to H,

we see that dn and dn" are injective at every point of H. Hence n , n

map neighborhoods Í7, , Í72 of rf in // diffeomorphically onto neighborhoods

Vx, V~2 of <lx, <l2 in C. Since ^oj = cb1on , if we let U = Ux n U2, and

replace VX,V2 by Vx n 0j~ (£/), K, n 4>2l(U), we conclude that 7r ,7t map

Í7 diffeomorphically onto VX,V2, and 0,(F,) = 02(K2). But this implies that

J^fcTf = Jh(cT2). So (£,,£,) is not in CrK(C, ,0, ;C2,02).This contradiction proves that CrK(C, ,0, ;C2,02) cannot contain a k-

dimensional Cw submanifold. D

Lemma 9.5. Let M be a Cw manifold, and let A e SBAN(M), dim A =

k. Then Clos^ = Ax uA2, where A2 e SBAN(M), Ax ç A, Ax is a k-

dimensional embedded Cw submanifold of M, A2 is closed, AxC\A2 = 0, and

dim^2 < k.

Proof. Let A = \J{A!: i e 1}, where {A1: i el} is a locally finite family of

K-dimensinal cubic images. Let (C¡, ß.,0.) be a realization of A'. It is

clear that ClosA = \J{4>fClosCi):ieI}. Let B¡. = B(Ci,4>¡). Then Bt isa

compact semianalytic subset of ß(.. Let ß. = CrK(C¡, (¡>¡ ; C ,0,), so F is a

semianalytic subset of ß( x ß , of dimension < k , by (A) of Lemma 9.4. Let

^2 be the union of all the 0((F,) and all the y/jj(B¡f , where yitfq , r) = 4>fq).

Then A2 e SBAN(M), A2 is closed of dimension < k , and A2 ç Clos A . (To

see that A2 is closed, let {xn} be a sequence of points in A2 that converges

to x e M. Assume x $ A2. Since the sets (pfBf are compact and form

a locally finite family, their union V is closed. So xn £ V if « is large

enough. Since x„ e ¿2, we must have x„ = <t>i(n)(qn) = <t>mirn)> for (<7„ >r„) e

Cr»f(c/(») ' 0/(n) ' C;(«) ' 0;(nj) • B? Passin8 to a subsequence, we may assume that

/(«) = i and 7(«) = j, independent of «. Also, we may assume that {qn}

converges to q and {rn} converges to r. Then x = 4>fq) = (¡>fr). Since x ^

/i2, we have (q,r) £ (B(C¡, 0,) xCfu (C, x B(Cj, 4>f). Therefore (q, r) e

CrK(C,, 0,, Cj ,0,), by (B) of Lemma 9.4. So x = ¥..{q, r) e w,fBij) ç A2.

We let Ax = A - A1. Then Ax satisfies the desired conditions. □

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REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS 451

Remark 9.1. In Lemma 9.5 is is not asserted that Ax is subanalytic. Naturally,

this conclusion will follow once we have proved Corollary 9.3. a

The proofs of Theorems 9.1 and 9.2 will be by induction on the dimension.

To carry out this induction, we will lift stratifications from R"~ to R" . The

relevant definitions are as follows.

If S is a block in M, and x: S —> R is a function, we use G(x) to denote

the graph of x, i.e. the set {(s, x(s)):s e S} . Also, we let G_(r) = {(s,t):s e S

and / < t(s)} , G+(x) = {(s ,t):s eS and t > x(s)} . If t , w are two functions

on S, write t < « if x(s) < t](s) for all s eS. In that case, define G(t , n) =

G+(r)nG_(tj).

An admissible function on the block S is a bounded, Cw function x on S

whose graph G(x) is a subanalytic subset of M x R. Clearly, if S is a block

and t is an admissible function on S, then G(x) is also a block. If t , w are

two admissible functions on S such that x < «, then G(x, «) is also a block,

because G(x, r¡) = f(E), where F = {(s, a, b, c): (s, a) e C7(t) , (s, b) e G(rj)

and a < c < b} , and f(s ,a,b ,c) = (s ,c). If S is a block, p e Clos S, and t

is an admissible function on S, then we use Ax(p) to denote the set of all limits

lim. ^ x(sf, for all sequences {s } of elements of 5 such that lim. ^ s = p .

It is clear that Ax(p) is a compact subset of R. If S ç A ç M, and t is an

admissible function on S, we call t A-regular if the limit lims t(s) exists—

i.e. if Ar(p) consists of a single point—for each p e An Clos S. We then use

Lt(p) to denote that limit. It is clear that Lz is continuous on A n Clos S.

Moreover, it is easy to verify that, if S is an yi-locally connected block, and

t an yl-regular admissible function on S, then G(x) is an (A x R)-locally

connected block. Similarly, if t , « are /i-regular admissible functions on 5,

and t < », then G(r,r¡) is an (A x R)-locally connected block.

Lemma 9.6. If S is an A-locally connected block in M, x an admissible function

on S, and p a point in A n Clos S, then Ax(p) is a compact interval.

Proof. We already know that Ax(p) is compact. Let a,b e Ax(p), a < b . Let

isi} > {°¡} be sequences of members of S such that lim,. s, = lim a =J J J 'OO J J KX) J

p, limj^oox(sf = a, \imJ^00x(af) = b. Let a < c < b. Let {Uk} be

a fundamental sequence of neighborhoods of p in M such that UknS is

connected for each k . Then x(Uk) is connected and contains points arbitrarily

close to a and to b. So c ex(Uk). Pick zk e Uk such that x(zk) = c. Then

lim^^ zk= p . So ce Ax(p). Hence Ax(p) is an interval. D

Corollary 9.7. If S is an A-locally connected block in M and x is an admissible

function on S such that (CIosG(t)) n (A x R) does not contain any nontrivial

vertical segment {p} x [a,b], a <b, then x is A-regular. a

An admissible system for a stratification S by blocks is a collection x =

{x :S e AT} , such that each x is a finite sequence (x{ , ... , xm,S)) of admis-

sible functions on 5 , with the property that t, < t2 < ■ ■ ■ < xm{S). (We require

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452 H. J. SUSSMANN

that m(S) > 0 for each S, i.e. we do not allow any of the sequences xs to

be empty.) In that case, we use G(x) to denote the union of the sets G(x^) ,

for 5 e AT and i— I, ... ,m(S). Also, we use ATZ to denote the set whose

members are all the sets G(xsf , S e AT, i = 1, ... , m(S), and all the sets

G(Ti .xi+\), for S e AT, i = I, ... ,m(S) - 1. It is clear that the members

of ATX are pairwise disjoint blocks. Notice that G(x) c \ATX\ c \AT\ x R and

n(G(x)) = n(\^\) = \<T\. If &> ç A, we call x /4-regular if every if is

,4-regular.

Lemma 9.8. Let AT be a stratification of a subset A of a Cw manifold M by A-

locally connected blocks. Let x bean admissible system for AT, such that G(x)

is relatively closed in A x R. Then x is A-regular and ATT is a stratification by

(A x R)-locally connected blocks.

Proof. We already know that the members of ATZ are pairwise disjoint blocks.

For each S, and each i e {I, ... ,m(S)}, the set (ClosG(tf)) n(^xR) is

contained in G(x), and therefore cannot contain a vertical segment. Hence t.

is /i-regular by Corollary 9.7. Therefore x is ^-regular. This implies that the

members of ATZ are A x R-locally connected blocks. One then easily verifies

that, whenever Se^, Te AT, T <z ClosS, F ¿ S, and ie{l, ... , m(S)} ,

there exists an index j = j(i,S, T) such that Lx¡ = t on F. It then follows

easily that the closure in A x R of a set G(x¡ ) is the union of G(x¡ ) and all

the G(xT](i s T)) for all the strata F e .AT such that F c Clos S and F ¿ S.

Also, if i < m(S), then the closure in A x R of a set G(x¡ , x¡+x) is the union

of G(rf,rf+1), G(x^), G(rf+]), together with all the G(x¡) for j(i,S,T)<

A: <7'(i+1,5,7'), and all the C7(t[,t[+1) for j(i,S,T) < k < j(i+ 1 ,S, T),

for all the strata F e <AT such that F c Clos S and F ¿ S. So ATX is a

stratification. D

Lemma 9.9. Let M be a Cw manifold. Let A, H eSBAN(M) besuchthat

H is closed of dimension h, and A ç H. Then there exists a finite family AT of

closed subanalytic subsets of M, such that every F eAT is a subset of Clos A

of dimension not greater than « - 1, with the property that, whenever S is a

connected subset of H which is compatible with AT, then S is compatible with

A.

Proof. We use induction on « and, for a given « , on k = dim A . Let P(h, k)

be the statement that our desired conclusion holds for a given h, k . We assume

that P(h',j) holds for all j if «' < «, and P(h,k') holds for k' < k.

Let A,H have dimensions k,h. Let B = Clos^. Using Lemma 9.5, write

B = C U D , where C ç A , C is a A:-dimensional embedded submanifold of

M ,D is a closed subanalytic subset of M of dimension < k , and CnD = 0.

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REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS 453

Also, write H = KuL, where K is an «-dimensional embedded submanifold

of M, L is a closed subanalytic subset of M of dimension < « , and K n L =

0.

Let F, = A n D, E2 = A n L. Let Hx = H, H2 = L. For i = 1,2, let

& be a finite collection of closed subanalytic subsets of M, such that every

G e & isa subset of Clos F., with the property that, whenever S ç H¡ and

S is connected and compatible with ^ , then S is compatible with F(. (The

existence of &x follows from P(h,k - 1) ,and that of A§2 from P(h - 1 ,k)A)

Let AT = %x u {B} u {D} if k < h , and / = g;u^2U {£>} U {L} if k = h .Clearly, every member of AT has dimension < « . Suppose 5" is a connected

subset of H, compatible with AT. Then S is compatible with ^ , and so 5"

is compatible with F, . Then either S ç Ex or S D F, = 0. In the former

case, S ç A. So we only need to consider the case when STiF, = 0. Since D e

AT, either S ç D or SnD = 0. If S CD, then SnA = SnDnA = SnEx = 0.So we may assume S C\D = 0. Assume first that k < h . Then B e AT. So

either S ç B or S n B = 0. In the latter case it is clear that S n A = 0.

If S ç B , then S = SnB = Sn (C U D) = (S nC)l) (S (lD)=SnC. SoS Ç C ç A . Now assume k = h . Then L e AT . So S is compatible with F .

If S ç L then, since S is compatible with A§2 (because AÀ?2 ç AT), we conclude

that S is compatible with Ex = A n L. So either S ç ,4 or 5ni = 0.

There remains to consider the case when 5flL = 0. Let K' = H - (Fu D),

C' = C - L . Then C', K' are embedded «-dimensional Cw submanifolds of

M, and C' ç K1. Since S ÇH, SnD = SnL = 0,we have S ç K'. It is

clear that C' is both open and closed in K'. So C' and K' - C' are unions of

connected components of K1. Since S is connected and S ç K', we conclude

that either S ç C' or S nC' = 0. In the former case, S ç A . In the latter

case, SnBCSn(C'uDuL) = 0, so SnA = 0. D

Call a subset sé of M Icb-stratifiable if, whenever j/ ç SBAN(M) is

locally finite, then there is a stratification ^5" of A by locally connected blocks

which is compatible with every member of sé . Clearly, such a set is necessarily

subanalytic.

Lemma 9.10. Let se ç SBAN(R"+ ) be a locally finite set of compact sets. Let

n be the projection (x,y) -» x from R"+1 to Rn. Let B e SBAN(Rn) be

closed. Let C e SBAN(Rn) be such that C ç B n n(\sé\), and n~X(C) n \sé\

does not contain a vertical segment {p} x [a,b], p e R", a < b. Assume B

is Icb-stratifiable. Let ^ ç SBAN(R") be locally finite. Then there exists a

stratification ^A of B by locally connected blocks, compatible with %f and C,

such that there exists an admissible system a on%A\C with the property that

G(o) = n~\C) n \sé\ and (VA\Cf is compatible with sé .

Proof. For each A e sé , let À = A n 7r~'(ClosC). Let ssT = {Ä:A e sé} .

It suffices to prove the conclusion with sé replaced by sé . Equivalently, we

may assume that \sé\ ç n~ (ClosC), so that n(\sé\) = ClosC. Since each

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454 H. J. SUSSMANN

fiber n' (c) n \sé\, for ce Clos C, is a compact subanalytic subset of the line

{c} xR, and this subset does not contain a nontrivial segment, it follows that the

fibers are finite. Therefore k = dimC = dim ClosC = dim 7r(|ja/|) = dim|j/|.

We use induction on k. The case k = 0 is trivial. So we assume that

0 < k < « and the desired conclusion is true for sets C' of dimension < k,

and locally finite families sé' of compact subanalytic subsets of R"+ such that

n(\sé'\) = ClosC'.We desingularize the members of sé . Precisely, we find compact Cw man-

ifolds NA, of dimension < k, and Cœ maps <&A-NA -* R"+1 , such that

®a(Na) = A . Let y/A = n o (p4. Let A^ be the set of those p e NA where y/A

has rank k, and let Na = NA- NgA . Then both A^ and A^ are semianalytic

subsets of A^ . We let A be the disjoint union of the A^ , and define <P, y/

to be the maps on A that agree with <P4, y/A on A^ . Also, we define N8 ,

A in an obvious way.

Let BA = y/A(NA), and BA = y/A(NA). Let A% be the set whose members

are the B8A and the BA . Then 38 is locally finite, AÁS ç SBAN(R"), and the

union of the members of AT! is Clos C.

Since B is Icb-stratifiable, there exists a stratification AT of B by locally

connected blocks, compatible with ^U/U{C}. Now let D = \ATk \C\. Then

D is an embedded Cw submanifold of R" and D e SBAN(Rn). Let Ñ =

y/~{(D). Then  is open in A, Ñ ç Ns , and ^[ is a proper submersion

onto D. (Indeed, if p e Ñ, then p cannot be in A because, if p e NA , then

y/(p) e BA , which is a set of dimension < k . So the stratum F of AT to which

p belongs would be contained in BA , and then dim T < k, contradicting the

hypothesis that y/(p) G D. Moreover, if p G A and lim. ^p = p , then

each y/(pj) belongs to a stratum F of AT. If it was not true that F = F for

sufficiently large j , then there would exist a subsequence {/7,(i)} such that all

the piU, belong to a T' e AT such that T1 =£ T. But dimF' < k, because

T' ç <p( A) ç Clos C . Since p e F n Clos F', we see that dim F < k , which

is a contradiction. Hence F = F for large enough j. So Pj e N for large

enough j. So Ñ is open in A, Â ç Ng, and y/ maps A onto D. Since D

is an embedded submanifold of R" , y/ is smooth as a map from  into D.

It is then clear that ^[A is a proper submersion onto D.)

If F e ATk \C (i.e. if F is one of the connected components of D), let

NT= y/~l(T). Then Âr is a A:-dimensional manifold, and y/\NT is a proper

submersion onto the ^-dimensional manifold F. So y/\NT is a covering map.

Since F is a block, and so in particular connected and simply connected, we

conclude that all the fibers y/~\t), t e T, have the same cardinality v(T),

and there exist v(T) Cw maps (f, ... ,Cj(r) from F to NT such that, for

each teT, y,~\t) = {rTx(t) ,C¡(t), ... XTm(Tft)}. Define rf (f) = i(Cf(0).where Ç is the projection (x, y) —> y .

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REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS 455

The sets NT are subanalytic in A, since they are inverse images of the suban-

alytic sets F under the analytic map y/ . Then the sets C, ( F) are subanalytic,

because they are the connected components of NT . Then the graph c7(t(. ) is

the image of if (F) under O, so it is subanalytic as well. For a fixed F, and

any pair of indices i,j in {1,..., m(F)}, let F . = {x e T: t(. (x) = t;. (x)} .

Then T¡ is subanalytic as well, because T¡ . = a(W¡ . f), where Wt. .. T =

(rJ(T)xrJ(T))n{(p,q)eNxN:<t>(p) = ®(q)}, and a(p ,q) = ip(p). Now

let AT' be a stratification of B by locally connected blocks, compatible with

AT and all the TtJ. Let & = AT'k\C. Let D' = \&\. Let f e &. Then

F' is contained in a stratum F of AT \C, and every set F(. is either dis-

joint from F' or entirely contained in it. This means that the functions xi ,

for 1 < / < v ( T), are such that, if two of them agree at one point of F',

then they agree throughout T1. Therefore, after appropriate relabelling, we

conclude that for each F' there is a finite sequence dx , ... ,ô ,T,. of Cw

functions on T1, such that àx < ô2 < ■ ■ ■ < à [T,., with the property that

$>(y/~l(T')) = G(ôxT')U---UG(à^T,)). The G(âf') are clearly subanalytic sets— I / T'

(because they are the connected components of n (T ) n\sé\), and the ojT' • * i

are bounded. Hence the di are admissible functions on F . Let à consist

of the à for all i and all T' e%A . Then è is an admissible system for ÎÀ.

We claim that ^ is compatible with sé . Indeed, let S e %Àa, A e sé .

Then either S = G(ôf ,àf+x) or S = G(ôJ ) for some F' e & and some i.

In the former case, S C\\sé\ = 0, so S C\ A = 0 . Now assume S = G(ôi ).

Let F e AT be such that t ç T. Assume S r\ A f, 0. Let z e S n A.

Then z = <P(/?) for some p e NA. Clearly, p e Cf(T) for some j, and thenT' ■ T i

<7; is the restriction of t to F . Since A^ is open in A, we conclude that

if (q) e NA for q e t, q near p. So «D(Cf\q)) e NA for q e t, q near

p . But then (q, ôf (q)) e A forqeT', q near p . So S n A is open in S.

Since A is compact, 5 n A is closed in S. Since S is connected, S ç A .

Let C = C — D . Then C is a subanalytic subset of B of dimension < k ,

because C' is the union of those strata of AT' \C that have dimension < k . Let

F = n~ ' (Clos C'). Let g7 be the family of all sets Ar\E, A esé . Then W is a

locally finite family of compact subanalytic subsets of Rn+ , of dimension < k .

Moreover, we have ClosC' = n(\W\), and n~\c')A)\W\ does not contain any

vertical segments. By the inductive hypothesis, there exists a stratification AT of

B by locally connected blocks, compatible with \AT'\, and an admissible system

à for %\C', suchthat G(ô) = n~l(C')n\W\ and (%\C')d is compatible with

g7. Now let a be defined as follows. If F 6 %A\C', we let oT = dT. If

F e %A\(C - C1), then let T' be the unique stratum of AT' that contains F, so

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456 H. J. SUSSMANN

that T1 e &, and ö is well defined, and then let a consist of the restrictionsT'

to F of the functions di . It is clear that a is an admissible system for ÍA\C.

Then ^, a satisfy all the desired conditions. D

We now prove Theorem 9.1. We want to prove that the following holds for

every m :

I (m): every subanalytic subset of a Cw manifold M is Icb-stratifiable.

We use induction on m . The case m = 0 is trivial. Assume I(m-l) holds,

and let M have dimension m. We want to prove that M is Icb-stratifiable.

One easily verifies that every locally finite union of Icb-stratifiable subsets is

Icb-stratifiable. Hence it suffices to show that a relatively compact subset of the

domain of a cubic coordinate chart is Icb-stratifiable. This means that we can

assume that M = R" and try to prove that a bounded closed cube K in Rm

is Icb-stratifiable. So we only need to consider finite families sé of subanalytic

subsets of K , and find, for each such family, a stratification AT of K by locally

connected blocks which is compatible with sé . In view of Lemma 9.9, we may

assume in addition that the members of sé are compact and have dimension

< m . Let ATA be the set of all closed faces of K of positive codimension. Let

sé = sé u 3AA . Then \sé\ isa compact subanalytic subset of Rw of dimension

< m.

By the Koopman-Brown theorem (cf. [KB]), there exists a regular direction

for \sé\, i.e. a nonzero v e Rm such that every straight line L parallel to

v intersects \sé\ in a finite or countable set. Since L n \sé | is compact and

subanalytic, it is in fact finite for every such L .

Make a linear change of coordinates so that v becomes the vertical vector

(0, ... , 0, 1 ). Write m = n + I, and use x,, ... , xn , y to denote the coordi-

nates. Also, write x = (xx, ... ,x ). After such a coordinate change, the cube

K need no longer be a cube with sides parallel to the axes, but we may find a

cube K' whose sides are parallel to the axes, such that K ç K'. Clearly, any

stratification of K' which is compatible with sé is compatible with both sé

and K . So the restriction of such a stratification to K will yield a stratification

of K compatible with sé , as desired.

Now let K' = K x [a, b], where A is a closed cube in R" and a < b . Then

\sé | ç K x [a, b]. Enlarge sé by adding to it the sets K x {a} and K x {b} .

This preserves the property that the members of sé are compact, subanalytic,

of dimension < « . Moreover, 7t(|j/|) is now equal to K, and \sé\ does not

contain vertical segments. By Lemma 9.10, with B = C = K and ß? = 0, we

see that there is a stratification ÍA of K by locally connected blocks, and an

admissible system o for %, such that G(o) = \sé\ and AT = "¥" is compatible

with sé .

Since G(o) is closed, we can conclude from Lemma 9.8 that AT isa stratifica-

tion by (K xR)-locally connected blocks. Since AxR is closed, the members of

AT are actually locally connected. Finally, it is clear that \AT\ = Kx[a, b] = K',

because K x {a} e sé , K x {b} ese , and se ç K x [a, b]. n

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REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS 457

Now that we have proved Theorem 9.1, it follows in particular that Corol-

lary 9.3 holds. So from now on we can use the fact that the class SBAN(M)

is closed under the operation of taking complements, as well as under finite

union and intersections. Since the class of subanalytic sets is also closed under

the operations of taking direct and inverse images by analytic maps—provided

that, in the case of the direct image of a set A, the map is proper on Clos A—,

one can in many cases prove easily that a set is subanalytic by just writing its

definition. Indeed, the logical connectives A ('and'), V ('or'), -> ('not'), cor-

respond to the operations of intersection, union and complementation of sets.

The existential quantifier 3 corresponds to taking a projection, and the uni-

versal quantifier V is expressible as -i3->. Hence any formula constructed by

means of these connectives from formulas that define subanalytic sets will still

define a subanalytic set, provided that the projections that correspond to the

quantifiers are proper. If AT = AT(xx, ... ,xn) is a formula with free variables

xx, ... ,xn, taking values in topological spaces Xx, ... ,Xn, and A§ is the for-

mula (Qxx)AT(xx, ... ,xn), where Q is one of the quantifiers 3 and V, we call

the quantifier 'Qx,' proper if, whenever K ç X2 x ■ ■ ■ x Xn is compact, there

exists a compact subset J of Xx such that (Qxx)AT(xx, ... ,xf) is equivalent

to (Qx{ e J)AT(x{, ... ,xn), for each (x2, ... ,xn) e K. The following result

then follows easily.

Theorem 9.11. Let Mx, ... , Mn be Cw manifolds, and let xx, ... ,xn be vari-

ables taking values in Mx, ... ,Mn. For j = I, ... ,r, let

Aj e SBAN(Mi(X JX x Mi(2j) x • • • x Mi{u(j)j)),

where each i(k ,j) is in {1,...,«}. Let AT be a formula constructed from the

atomic formulae '(x/(1 ,),••• >x¡tV(¡\ ,f € A. A by means of the logical connec-

tives A, V, -i, 3, V. Assume that all the quantifications are proper. Let AT have

x( , ... ,x( as free variables. Then the set A of those s-tuples (x(. , ... ,x. ) for

which AT holds is subanalytic in M, x ■ ■■ x M . a

We now prove Theorem 9.2. We first consider the special case when M, N

are Euclidean spaces and / is a projection. If p > q , we use np to denote the

projection (x,, ... ,xp) -► (xx, ... ,xq).

We assume that

I. L is closed subanaltyic in R"+m t

II. / = n^+m , A = f(L), dim A = k , and fis proper on L,

III. sé is a locally finite family of subanalytic subsets of Rn+m ,

IV. 3§ is a locally finite family of subanalytic subsets of Rn .

We first prove that I(k,m,«) holds for every m,« and every k e {0, ... ,

«}, where I(k,m,n) is the assertion that, whenever f ,L,A,sé ,3§ satisfy

conditions I, ... , IV, then there is a closed subanalytic subset A0 of A, such

that dimA0 < k and stratifications AT, AT of L+ = F-/"'(A0),A, respec-

tively, such that, if A+ - A-A0 , and L0 = /~'(A0)nF, then AT is compatible

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458 H. J. SUSSMANN

with A2§ U {A0} , AT is compatible with sé ,AT consists of locally connected

blocks, and AT consists of (A+ x Rm)-locally connected blocks, and is a block-

lift of 9~ by /.We prove this by induction on m, for fixed « and k. Obviously, the

case m = 0 is just a corollary of Theorem 9.1. Assume that m > 0 and

I(k,m - 1,«) holds. We want to prove that I(k,m,n) holds.

Assume that f,L,A,sé ,SA% satisfy conditions I, ... , IV. Let k = dimF,

so that k < k + m . Let / bea locally finite family of compact subanalytic

subsets of R" (e.g. cubes) whose union is R". Let sé consist of all the sets

Anf~l(J)nL,for A ese , J eS. For each A e sé , let sé" (A) be a finite

family of compact subanalytic subsets of Clos A , of dimension < k , with the

property that, if «5^ is a stratification such that \AT\ ç F and AT is compatible

with sé (A), then it follows that AT is compatible with A . (The existence of

sé (A) is assured by Lemma 9.9.) Let sé = \J jSé(A). Then sé is a locally

finite family of compact subanalytic subsets of F of dimension < k , such that

every stratification AT, compatible with sé , for which \AT\ ç L, is necessarily

compatible with sé .

Let K = \sé |, so K is a closed subanalytic subset of F, and dim K < k .

Write Rn+m — r" x Rm, with coordinates (x,y), x = (xx, ... ,xn), y =

{yx, • • • Tm) ■ For each (x ,y), and each vector v e Rm , let Xx v be the

straight line {(x ,y + tv): t e R}. Let B(v , K) be the set of those x e A with

the property that, for some y, the set Xx v n K is infinite. Then B(v ,K) e

SBAN(R") by Theorem 9.11. (Indeed, if x e Rn, then x e B(v,K) if and

only if

(3y)(3t)(3x)((y,v) = OAt < x /\(Vs)(t < s < x => (x,y + sv) e K)).

Since / is proper on K , the quantifiers in the above formula are proper.) Call

v regular if v / 0 and dim B(v ,K)<k.

We show that a regular v exists. To see this, let G be the set of those

(x,y,v,t,x) e R'xfxfxRxR such that (y, v) = 0, t < x, and

(x ,y + sv) e K for t < s < x . For each x, v , let H(x, v) be the set of those

(y,t,x) such that (x,y,v , t,x) e G. Let F be the set of those (x,v) for

which ||u|| = 1 and H(x ,v) ^0. Then E,G are subanalytic in Rn+m , Rn x

R'" x Rm x R x R, respectively. (This follows from Theorem 9.11, by just writing

down explicitly the definitions of F and G.) Let r\,p,v be the projections

(x ,y ,v ,t,x) —> (x ,v),(x ,v) —> x, (x ,v) —> v , respectively. Then t](G) = E .

We claim that dimF < k + m-l . Indeed, by Theorem 9.1, F is a locally finite

union of blocks. If dim E > k + m - I , then it follows in particular that F

contains a block S of dimension k+m-l . On the other hand, Gs = Gn«~ (S)

is a finite union of blocks. So Gs contains a block S' of dimension k + m-l

that projects diffeomorphically by « onto 5. So there is a Cw map y/ from

S into G such that « o y/ = identity. Let p , v' be the restrictions of p, v to

S. Then p , v' are maps into A and the unit sphere I of Rm , respectively, so

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REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS 459

their ranks cannot exceed k, m - 1. So the ranks actually are equal to k and

m - 1. This means that, after shrinking S, if necessary, we may assume that

S = Uxíl, where U isa A:-dimensional Cw submanifold of R" suchthat U ç

A, and ß is open in I. Now write y/(x ,v) = (x ,y(x ,v) ,v ,t(x ,v),x(x ,v)).

Let V = S x R, V = {(x,v,s):(x,v) e S,t(x,v) < s < x(x,v)}. Let

0(x,v ,s) = (x,y(x,v) + sv) for (x,v ,s) e V. Then V is a submanifold of

R" x I x R of dimension k + m , V is open in V, and 0 maps V into K.

Clearly, 0 has rank k + m at (x,v ,s) if s is large enough. Since 0 is Cw and

V is open in V, we conclude that 0 has rank k + m at some poiint of V . But

this is a contradiction, because 0 maps V into K, and dim K = k < k + m.

So dim F < k + m - 1 as stated. Now, it is clear that x e B(v, K) if and only

if (x, v) e E. If there did not exist a regular v , it would follow that every set

v~\v)f)E, v e X, is k-dimensional. But then dimF = k + m - 1, which is

contradiction. So a regular v exists.

Now pick a regular dgR™. After a linear change of coordinates in R'" ,

we may assume that v = (0, ... ,0,1). Now write Rm = Rm_1 x R, with

coordinates (z,t), z = (zx, ... ,zm_x), zteR, t e R. Let £,£ be the

projections (x,z,t) —> (x,z),(x,z) —> x, respectively, so that / = Ç o Ç.

Let T = í(L). Let Ä0 = C1osF(?j,A). Let f0 = C"'(À0) nl\ Let FQ =

<r'(f0) n F . Let L+ = L-L0, f+ = Y - TQ , Ä+ = A - Ä0 . If (x, z) e f+ ,then the vertical line {(x,z)}xR interesects K in a finite set. Hence by Lemma

9.10 there exists a stratification ¿f of T by locally connected blocks, compatible

with f+ , and an admissible system ö for S[f+ , such that G(o) = ^_1(r+)nA

and S" is compatible with sé .

Next we apply the inductive hypothesis that I(k, m - 1, «) holds. We obtain

a closed subanalytic subset A0 of A, of dimension < k, a stratification AT

of A by locally connected blocks, compatible with Ai% u {A+} u {A0} , and a

stratification ^ of f+ by blocks—where f0 = C~'(A0)nr, A+ = A-A0,

and f+ = T - f0 = C~'(Â+) n T— compatible with Q U {f+} , that consists

of (A+ x Rm~ )-locally connected blocks and is a block-lift of AT by C • Let

A0 = À0UÂ0, r0 = f0uf0, F0 = Z0UF0, A+ = A-A0, T+ = T - T0 ,

L+ = L - F0. Then A0 is a closed subanalytic subset of A, of dimension

< k. Since AT is compatible with A+ , the set A+ is a union of strata of

AT . Since V is a block-lift of AT by Ç, every inverse image Ç~'(F) n T, for

F e AT\A+ , is a union of strata of % that are (A+ x Rm~')-locally connected

and are block-lifts of F by Ç. In particular, this is true for those strata of AT

that belong to AT\A+ . Let %A' = %\T+ . If U eW , then U is (A+ x R'""1)-

locally connected, because A+ ç A+ , and U = <S>U(CU x T), where Cu is an

open cube in a Euclidean space of the appropriate dimension, and <PV is a CM

diffeomorphism on Cv xT with subanalytic graph, such that <Pv(u,v) = v

whenever u e Cv, v e T. On the other hand, U is contained in a unique

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460 H. J. SUSSMANN

stratum QL, of €. Since U Q T+ ç f+ , we see that Qv e S\Y+, so the

functions a¡ are defined on Qv . Let a be the system of functions obtained

by restricting the of1' to U. The collection o of all the au , for all U e %A',

is an admissible system for %A'. Moreover, it is clear that G(a) = G(ö) n F+ =

AnF+ , and \%A'a\ is compatible with séu{L+}, and therefore with séu{5?+}.

On the other hand, %A' is a stratification of T+ by A+xRm_1-locally connected

blocks, and the graph G(o) is closed in F+ , which is closed in A+ xRm . So, by

Lemma 9.8, %A'a is a stratification by (A+x Rm)-locally connected blocks. It is

clear that \%A'a\ = L+ . Hence, if we take AT = f/'" , all the desired conditions

are satisfied.

We now prove that J(k,m,n) hold for all n,m,k, where J(k,m,n) is

the statement that, whenever I, ... , IV holds, then there exist stratifications

AT, AT, of R"+m t r" i respectively, such that AT is compatible with sé u {L} ,

AT is compatible with AS u {A}, AT consists of locally connected blocks, AT

consists of blocks, and AT\L is a block-lift of AT by /. The proof is by

induction on k , for fixed « and m . The case k = 0 is trivial.

Assume J(k - 1 ,m,n) holds. Let fi ,L,A,sé ,A3§ satisfy I,..., IV. Pick

stratifications AT, AT of Rn+m t R" by locally connected blocks, compatible

with sé U {L}, <% U {A}, respectively.

Using I(k,m,n), find a closed subanalytic subset A0 of A, such that

dimA0 < k and stratifications AT, AT of L+ = L - f~ (A0), A, respec-

tively, such that, if A+ = A - AQ, and LQ = f~ (AQ) n L, then AT is

compatible with AT U {A0}, AT is compatible with AT, AT consists of lo-

cally connected blocks, and AT consists of blocks and is a block-lift of AT by

/. Next, using J(k — l,m,n), find stratifications AT, AT of Rn+m , R" ,

compatible with AT U AT u {F0} , AT~u AT' u {A0} , respectively, such that AT

consists of locally connected blocks, AT consists of blocks, and AT\LQ is a

block-lift of AT by f. Now let SP = (AT\L0) U (ÂT\L+) U (ÁT\(Rn+m - L)),

AT = (AT\A0) U (f^\A+) U (AT\(Rn - A)). Then S?,AT satisfy all the desiredrequirements.

This completes the proof of Theorem 9.2 for the special case when M, A

are Euclidean spaces and / is a projection. The general case can be deduced

from this by using the fact that every real analytic manifold has a proper real

analytic embedding in a Euclidean space (Grauert's theorem). Alternatively,

one can use the much simpler fact that a real analytic manifold has a proper

C subanaltyic embedding in a Euclidean space (cf. Hardt [H3, p. 215]). d

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License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

REAL-ANALYTIC DESINGULARIZATION AND SUBANALYTIC SETS 461

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Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

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