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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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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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