IC/95/3

INTERIMATIOIMAL CENTRE FOR

THEORETICAL PHYSICS

DEFORMATION RETRACTS OF STEIN SPACES

Helmut Hamm

and

Nicolae Mihalache

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

MIRAMARE-TRIESTE

IC/95/3

International Atomic Energy Agencyand

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

DEFORMATION RETRACTS OF STEIN SPACES

Helmut HammMathematisches Institut der Westfalische Wilhelms Universitat Miinster,

Einsteinstrasse 62, D-4400 Miinster, Germany

and

Nicolae Mihalache1

International Centre for Theoretical Physics, Trieste, Italy.

MIRAMARE - TRIESTE

January 1995

1 Permanent address: Institute of Mathematics of the Romanian Academy. P.O. Box1-764, 70700 Bucharest, Romania.

1. Introduction

Let X be an n— dimensional Stein space. It was proved in [4, 5, 3] that X has thehomotopy type of a CW- complex of dimension < n; in the algebraic case this wasproved in [8] with the additional conclusion that the CW-complex is finite. It istherefore natural to ask if there exists a subset Q of X with the same topologicalproperties as X, for instance Q is a strong deformation retract of X, and Q is aCW-complex of dimension < n. The main result of the paper is:

Theorem 1.1 Let X be an n— dimensional Stein space. Then there exists Q a closedsubanalytic subset of X ,d\mQ < n such that Q is a strong deformation retract of X.If X is affine algebraic then Q can be chosen to be compact.

Above we give to X the subanalytic structure induced considering it as a closedanalytic subset of some C^ via the embedding theorem of Remmert.

The result is not trivial since we are considering complex dimension for X andreal dimension for Q. The results cited above are consequences of Theorem 1.1 .

One of the ingredients of the proof of the main result is what we call embeddedMorse theory. This consists in finding inside a real analytic manifold with boundaryM a closed subanalytic subset which is a strong deformation retract of M and whosedimension (usually segnificantly lower than dimR M) can be computed as a functionof the indicies of the critical points of a Morse function on M.

It would be interesting to write down such a theory for stratified spaces but thereare some technical difficulties. We mention a recent advance made by Coste andShiota [1] which proved a semialgebraic version of Thorn's first isotopy lemma (thesubanalytic case will appear in a forthcoming book of Shiota [14]). Since the proofof the main result is by induction it turned out that a slightly more general result istrue:

Theorem 1.2 Let X be an n— dimensional Stein space and P be a closed subana-lytic subset of X. Then there exists Q a closed subanalytic set, Q D P, dimQ <

dimP + l ,n} such that Q is a strong deformation retract of X.If X is affine algebraic and P is compact then Q can be chosen to be compact.

In fact a relative form of the result (for a pair (A", A) where A is a (closed) analyticsubset of the Stein space X) is true (see Corollary 4.3). This comes close to provinga conjecture of Goresky-MacPherson ([3], page 152). It is an easy consequence of thetriangulation theorem for subanalytic sets that in Theorems 1.1 and 1.2 the homotopy(which shows that the corresponding sets are strong deformation retracts of X) canbe chosen to be subanalytic i.e. his graph is subanalytic.

The second key ingredient of the proof is Shiota's linearization theorem for sub-analytic functions. In particular it is not a luxury asking in the main result that theclosed set Q is subanalytic rather than a CW complex.

The paper is divided into three sections. In the Preliminaries we develop a toolneeded in the embedded Morse theory. We introduce the "n—trail property" for apair (B,A) of closed subanalytic sets in a real analytic manifold. For n = — 1, the( — 1)— trail property (therefore called simply: trail property) for a pair (B,A) impliesthat A is a strong deformation retract of B. The notion is inspired by the followingfact from PL-topology ([11], page 40): "Let {B,A) be a compact polyhedral pair insome Euclidian space. Suppose B \ , A where \ , means collapsing. Then given P C Ba closed subpolyhedron there exists Q C B a closed subpolyhedron, Q D P (called thetrail of P under the collapse) such that dimQ < d imP + 1 and B \ AUQ". Theorem1.2 says exactly that the pair (X, 0) has n—trail property if X is an n—dimensionalStein space. In the second Section we prove the basic facts concerning embeddedMorse theory and in the third Section we prove Theorem 1.2. A first form of thepaper was circulated as a preprint [6]. In the first part of that preprint a differentproof was presented for the affine algebraic case of the Theorem 1.1 with the strongerconclusion that the compact Q (in the notations of the Theorem 1.1) is semialgebraic(that proof was in the spirit of the Le's construction of a "polyedre evanescent" of anisolated singularity, see [9]). The fact that Q can be chosen semialgebraic in Theorem1.1 can also be obtained using the method of proof of this paper but developing avariant of embedded Morse theory for Nash manifolds with boundary and using thepiecewise linearization theorem for semialgebraic functions (see [2, 13]).The second author acknowledges support from Deutsche Forschung Gemeinshaft andInternational Center for Theoretical Physics during the preparation of this work,

2. Preliminaries

2.1

Without loss of generality we will consider all Stein spaces embedded as closed analyticsubsets in some complex Euclidian space. For basic facts about subanalytic sets wesend to [7]. A function / : A i-> R where A C RW is subanalytic if it is continuousand his graph is subanalytic (when viewed as a subset of RN x R ). Since subanalyticsets are triangulable (see for instance the references in [12]) these sets are particularlycomfortable to work with. From classical Algebraic Topology results ([15]) it follows

that if A A- B are closed subanalytic subsets then A is a strong deformation retractof B iff i is a weak homotopy equivalence. In particular A is a strong deformationretract of B iff A is a deformation retract of B. Also if A is a deformation retract of

B which is a deformation retract of C then A is a deformation retract of C.We will use in the paper Morse theory on manifolds with boundary as developed

in [4]. Recall only that if M is such a manifold and ip : M >—>• [0 oo) is a definingfunction for the boundary dM of M then a critical point p of x\9M (where x 1S a

real valued function on M) is of type / (respectively E) if dPx = Xdpip with A > 0(respectively A < 0).

2.2 n— trail property

If X is a (strong) deformation retract of Y and Z n Y C X then clearly X U Z is adeformation retract of Y U Z.Applying two times this fact (which we shall denote (*)) we obtain:

Lemma 2.1 Let Z\,Z2 be two closed subanalytic subsets in a real analytic manifold,C = Z\ fl Z2. Let Aj C Z} (j = 1,2) be closed subanalytic sets such that Aj U C is adeformation retract of Zi, A\ C\C C A2 and A2 is a deformation retract of Z2. ThenAi U A2 is a deformation retract of Z\ U2 2 ,

Corollary 2.2 In the same situation as above if instead of Ai D C C A2 we supposethere exists a closed subanalytic subset D such that A\ D C C D and A2 H C is adeformation retract of D then A\ U D U A2 is a deformation retract of Z\ U Z2.

Proof of 2.2. Since by hypothesis A2 C\ C is a deformation retract of D from (*)it follows that A2 is a deformation retract of A2 U D. Therefore we may apply theLemma for Ai and A'2 = A2 U D.

L e m m a 2.3 Let X be a finite cellular complex (in some Euclidian space) and P C Xbe an n—dimensional polyhedron. Then there exists Q C X an < (n-\-l)— dimensionalpolyhedron such that P U Xn C Q and Xn is a deformation retract of Q.

Above Xk is the k— dimensional skeleton of X.Proof. Let l(P) - min{A;/P C Xk}. We make induction on / = l(P) with theadditional hypothesis that Q C X1. For / = n is clear. Let P C X be a polyhedronwith/(P) = 1 + 1(1 >n). For each (/ + 1) cell C of X consider bc C Int (C) an (/ + 1)(open) PL ball such that bcC\P = 0. Then X'+1 \{jbc \ X' were \ means collapsingin the sense of PL topology, see [11], page 40. Let Q\ be the trail of P under thiscollapse. Then P C Qi C Xl+\ dimQj < n + 1, dim(Qi f l X ' ) < n and Qi n X1 isa (strong) deformation retract of Q\. By the induction hypothesis let Q2 C X1 suchthat Q2 D Q\ H X1, Qi D Xn and Xn is a deformation retract of Q2. Then we maytake Q = Qi U Q2 as the polyhedron satisfying the conclusions of the Lemma. I

Let A C B be closed subanalytic subsets of a real analytic manifold.

Definition 2.4 (B,A) has n— trail property if given P C B a closed subanalyticset there exists Q C B closed subanalytic such that Q D P, d'imQ < max{dim P +1, n}, d\m(Q n A) < max{dim P, n — 1} anc/ AU Q is a deformation retract of B.

Here dim0 = — oo. Note that if (B,A) has n—trail property then it has n— trailproperty for any m > n. Also that if (B,A) has ( — 1)—trail property then A is adeformation retract of B (but not the converse !). We will call "( — 1)— trail property"simply "trail property".

Definition 2.5 {B.A) has exact n—trail property with respect to C C B if givenP C B we find Q as in the definition 2Jf and moreover Q D C — P D C.

Arguing like in the second part of the proof of Lemma 2.3 we prove:

Lemma 2.6 / / A C B C C are dosed subanalytic sets of some real analytic manifoldand if D C C then if(B,A) and (C,B) have exact n—trail property with respect toD fl B respectively D then (C, A) have exact n— trail property with respect to D.

Of course, a particular case of Lemma 2.6 is when D = 0 in which case we speakabout n—trail property for (B, A), (C,B) and (C,A). As above "exact ( — 1)—trailproperty" will be abbreviated with "exact trail property".

Lemma 2.7 Let X be a closed subanalytic noncompact subset of some real analyticmanifold and ip : X H-> (0, oo) a subanalytic exhaustion function. Let n £ Z . LetRv f~ oo (RQ — 0) be such that for each u (tp~l([R1/, Rv+\\), il>~1({Rl,})) has exactn— trail property with respect to ip~l({R!/+i}).

Then (X, 0) has n— trail property.

Proof. Let P C X be a closed subanalytic subset and k = max{dim P, n}. Let Sv bethe k— dimensional skeleton of a subanalytic triangulation of tp~l({Rv}) compatiblewith P 0 V ~ 1 ( { ^ » . For any v let Qv D S^+i V {P D ip~l([Rv, R*+i])) be a closedsubanalytic subset of ^"'([fl,,, Rv+i]) such that Qu D ip~1({Rl/+i}) = Su+U dimQv <k + l,dim(Q^ PI tp~l({Rl/})) < k and ^1{{RJ)) U Qu is a deformation retract of

Next for any v choose (according to Lemma 2.3) Dv C V7 1({-^"}) cl°se<i subana-lytic such that A, D (Qur\i})~x{{Ri,}))yjSSJ, dimZ?^ < k + 1 and Sv is a deformationretract of Du, Let Q — [JiQt, U Du). Since ip is an exhaustion function it is enoughto check that Q Pi {ip < Rt,} is a deformation retract of {^ < R»} for any v and thisfollow by induction on v using Corollary 2.2. I

Remark 2.8 With X and ij' as above if there exists 0 < RQ < ... < R»o < oosuch that {i})~1([Rv, Rv+i]),iJ>~l({Ri,})) has n— trail property for u < v0 — 1 while({•0 > R^j.ip'^lR^})) has n— trail property then again (Jf, 0) has n— trail property.Also if X is compact ^ : X H (0, oo) is subanalytic and there exists 0 < RQ < ... < R^such thatiJt(X) C (0, R^) and (0" 1 ([/?„, Ru+i]),tff-l({Rv})) has n-trail property forv < UQ — 1 then (X, 0) has n—trail property.

Both assertions follow from Lemma 2.6 (with D = 0) rather than Corollary 2.2.

Lemma 2.9 Let C C A C B be compact subanalytic sets in a real analytic manifold.Suppose h : A x / i-> B is a subanalytic map such that /i(a, 0) = a Va € A, h(a,t) =aVae C, B = h{(A \ C) x /) U C w/*ere fc((A \ C) x /) D C = 0 anrf h\(A\C)xi is ahomeomorphism on image.

Then (B,h(A x {1}) has exact trail property with respect to A.

Proof. Define H : B x / K» B by H(y,s) = h(x,s + (1 - s)i) for y = h(x,t), x GA \ C, i 6 / and H(y, s) — y for y G C, * € /- It is easy to see that H is continuous(and subanalytic). Using // we see also that h(A x {1}) is a deformation retract ofB.

Let now Q = {h(x,s)/Bu < s such that h(x,u) € P} C B. Then Q is closed sub-analytic (it can be described as h{SH{h~l{P))) where SH{h-\P)) = {(x,s)/3u < ssuch that (x,u) G / ^ ( P ) } is obviously subanalytic). We have H(Q x /) C Q and us-ing this H it follows that QC\h(A x {1}) is a deformation retract of Q so Qu/i(/4x{l})is a deformation retract of B. The rest is straightforward. I

3. Embedded Morse theory on manifolds with boundary

Let M' be a real analytic manifold (without boundary) and M C M' be a realanalytic manifold of the same dimension possibly with boundary. If dM = 0 wesuppose M = M'. If the boundary of M is not empty we will suppose dM = NQ U JVIwhere iV0, Ni are disjoint open subsets of dM. Suppose also that ift is a real analyticfunction on M' defining dM such that M = {tp > 0).

Let x be a real analytic function on M'.

Theorem 3.1 Let a < (3 be two regular values for x and x\3M such that x~l([M is compact and, defining x = X|x-i(K/?])nMj X *s an rn — function with respect toX~l([<x,0])- Suppose there exists n G Z such that x an^ X\NO have critical pointsrespectively critical points of type I of index < n while x\Ni has critical points of typeE and index < n — 1; moreover that to a critical value ofx or X\OM corresponds onlya critical point.

Then (JT1([a,/?]).F'1({o}) u (^i n X"1^/?]))) has exact n~ trail property withrespect to x"1 ({/?}) U (No D ^([a, [3])).

Proof. Step I. Suppose first that x~ doesn't have critical points, x\Na doesn't havecritical points of type / and XIJV, doesn't have critical points of type E.

Arguing like in [4], Proposition 1 we find first a C°° vectorfield v on a neighbour-hood ofx'HfaP]) i n M' s u c h t h a t (O^xt") = - 1 o n X"1 ([",/?]), (ii)diff(v) > 0 onNo^X~l([a^}) anc i {iu)dijj(v) < Oon jVinx-l([a,^]). We approximate v in theC 2 -topology with w real analytic satisfying (i')dx(w) < 0 on X 1 ([^i fl]) an fl (^)i (**0-Let a be the flux of w (<r(q,0) = q). For q G x~Xi\Q->0\) define rg = sup{r > 0/a(q,-)is defined on [0, r) and a(q, [0, r)) C x"~'([Q^])}- (*')> (")i (n«) and elementary prop-erties of the trajectories of vectorfields imply that rq exists (and is < oo), a(q, •) is de-fined on [0,r,] and<r(?,r,) G x~1({a})U(/V1rix"I([o,/?]))- Moreover x"1 ([«»/']) 3 ^r, is semi-analytic (in particular continuous) since locally it is the maximum of tworeal analytic functions. Let h : (x~'({/?})U( Wx-l ( [a , /? ] ) ) )x / M- x~l{[<x,@]) be de-fined by h(q,t) — cr(q, trq). It is easy to see that h satisfies the conditions of Lemma 2.9(with C - Nonirl{{P}) thus proving that {!T\W^]),irK{<A)^{^^irl{[^P\))has exact trail property with respect to x"1 ({/?}) u (^o ( I x ' ^ a , ^ ] ) in this case.Step If. Using Step I and Lemma 2.6 we may suppose that x or X\dM has a uniquecritical point in X~l([aiP}) which is of type / if p G iVo or E if p G M, that x(p) = 0and a = —f, ^ = £ where £ > 0 is arbitrary small.

Let U be an open neighbourhood of p in M' on which there exists a real analyticsystem of coordinates centered at p, (uu. .. ,um) whose restriction to U d M is asystem of coordinates on M and such that:

X\u = ~£ + t] if p $. dM respectivelyX\u = -£ + ?}' + $um if p € OM

where £ = u\ + . . . + u\, r) = uj|+1 + . .. + u2m, rf = u2

x+l + . .. + < _ : , A is the indexof p, <S = 1 if p is of type / and 8 = - 1 if p is of type £" (see [4], page 128).

Let ^ : ( / f l iWi- f [0 ,oo) be defined by /Y = £ + ?/ and let <5 -C 1 such that {/i < «5}is compact. We may suppose e « l such that X\{^=s} has no critical points in [—e,s].Define finally x = X\{^<S}-

Using once more Step I (with No = 0) and Lemma 2.6 it is enough to seethat (x~l([—e,e\), X~l({£})) has exact n— trail property with respect to {fx~l{{8}) D

l

We will analyze the three possible cases: p ^ <9Af, p £ No or p £ Ni and will showin each case using Lemma 2.9 that (x~L([-£i £]), X~1({~£}) u Cell) has exact trailproperty with respect to (^-1({^}) n X~'(["«) £])) U X^Hi^}) w e r e Cell is exactly thecell adjoined when crossing the critical level set {x = 0}.Case A p £ dM. We are in the situation of the classical Morse theory. Here Cell=X-1([—£, £])n{?7 = 0} is a closed subanalytic set of dimension A < n. Let a : {pi < 8} xI i-> {(* < £} be defined by <r{q,t) = (u^q).. .ux(q),(l - t)ux+1(q},... (1 - t)um{q))

(this is nothing but the flux of grad r) where grad is taken with respect to the metric onU induced by the Euclidian one). For q € (n~l({8}) D x~l([-£>£])) u X^ii6}) definerq = supM^u) G r1([-e,e])V« G [0,01- Then af^r,) € (^"'([-e^D n iv =0}) U ,Y~1f{~£}) and q i—)• r9 is semi-analytic (one can even write down explicitformulas for r,). Defining h : ((^(W)nr'([-^]))urH(4)) x / ^ rHh^e])by /j(f/, f) = a{q, trq) we easily see that we are on the hypothesis of Lemma 2.9 .Case B p G JV0 (and ao p is of type / ) . The proof of Theorem 5 in [4] shows that inthis case Cell is still x-1([—£,&]) f~l {// = 0} which is closed subanalytic of dimensionA < n. We use the same a : {fi < 6} x / H> {/i < 6} to produce /i : ( (^^({^ j ) Pi

X"'([-£,£])U{x = e } ) x i ^ x -1[—£.£] w i t h Im/ii = ^ " ' ( { -^DU C e l 1 a n d satisfyingthe hypothesis of Lemma 2.9 .Case C p € Afi (and so p is of type E). The cell adjoined when crossing the level set{x = 0} is X~l([—£, £}) H {??' = 0} which is closed subanalytic of dimension A + 1 < n.We use now a' : {ji < 6} x / >-> {fi < 8} given by a'(q, t) = (ui(«y), . . . ,u\(q), (1 —t)u\+1{q),. . . ,(1 - *)«„,_,((?),«„,(<?)) to construct /t : ((/i~1({6}) n x~ l ( [ - £ i £ l ) ) u

Y~!({£})) x ^ ^ ^ " ' ( h ^ ] ) s u c h that ImAi = x ^ d - e ^ U C e l l and satisfying theconditions of Lemma 2.9 . I

Using Lemma 2.7 (and Theorem 3 in [4]) from Theorem 3.1 follows quickly:

Theorem 3.2 Suppose x — X\M is on m— function and there exists an n such thatX (respectively X\Na) ^

as cr^ca^ points (respectively critical points of type I) of index< n and XiN} has critical points of type E of index < n — 1.

Then (M, N\) has n— trail property.

Remark 3.3 If M is compact or if there exists R such that. \ respectively X\N0 respec-tively Yiyv have no critical points respectively no critical points of type I respectivelyno critical points of type E on {\ > R} then using Lemma 2.6 we may prove that(M, N\) has exact n— trail property with respect to No.

Corollary 3.4 Let X be a q— complete (q = 0 meaning Stein) n— dimensional com-plex manifold. Then (X, 0) has (n -f q)— trail property.

Of course for q — 0 this gives the main result in the smooth case.

4. Proof of the main result

We may suppose that X is embedded in some C^. Using Lemma 2.6 and inductionon n — d imX it is enough to prove that given A 3 Sing (A") an analytic subset of Xthen (X,A) has n— trail property.

Fix E a complex analytic Whitney regular stratification of X compatible withSingfX) and let A be the union of strata of dimension < n — 1. Let xp : X i-> (0, oo) bea real analytic strongly plurisubharmonic exhaustion function and / J , , / o o (RQ = 0)regular values for t/> (with respect to E). Let also ip : X t—>• [0, oo) be real analytic suchthat A = {ip = 0}, logip is plurisubharmonic and <p\x\A ls strongly plurisubharmonic(see for instance [10]). Let S = {(£„)/£(, \ 0} on which we introduce the relatione < rf if ev < nu+l W . For each e G S define VE = U { ^ < V-1 < Rv+i ,H> < £»},V+ = {J{RV < i> < R,,+l,<p > £„} and dV£ - Ve D V+.

It is enough to show that:(A) there exists e° € <S such that for e < e° (V+,dVe) has n— trail property.(B) there exists £] € S such that for e < £l (Vr, A) has trail property.Proof of (A) Using the curve selection lemma there exists V neighbourhood of A inX such that:(1) <f and <p\{y,=Rv) doesn't have critical points on V \ A

For each u define (like in [5], page 2) \ v : ̂ ~l([R,y, Ru+2}) h^ [0, 1) by:

0 on AH

where Mv > ip on V~ L ( [^ , ^+2]) . Xi/ ' s continuous and real analytic outside A.Choose for any v av > 0 such that:(2) \i' a n d X\^=R,, doesn't have critical values in (0,0/^] V^ ([5], Lemma 2).

Now let £° G 5 be such that V,o C V, {Rv < $ < R,,+i,tp < £°} C {Ru < $ <Rv+i,Xi/ < Q f} Vi/ and moreover:(3) there is no relation d\u = —Xdf with A > 0 on {/?„ < ip < R»+i ,0 < tp < £°} Vi/([10], Lemma 3.4 (Hi)).

Fix e < s°. Using Lemma 2.6 it is enough to show:(E) (K+ n 4^1{[R, R]),dV€ n ip-\[R\ R])) has exact n - t r a i l property with respectto V+n%h-]({R}).

Above R' = Rv R — R^+i. In turn (E) splits into:

(E-[) ({ti> < R, x > Q}J { '̂ < R'. X = a}) has exact n — trail property with respect to{^j = R, Y > Q} and

(£2) ({/?' < V < #,v? > £,* < « } , { ^ - /?',*> > £} U {i?' < ^ < R,*J> = £}) hasexact trail property with respect to {ip = R,<p > £, \ < a}).

Above x = Xv> e = eu, a = av.Given the way /?, R', e where chosen C — {R! < rp < R, ?/; > e} is a manifold with

corners and the proof of (E2) is completely similar to Step I in the proof of Theorem3.1. We omit the details. To prove (E\) let x = X\c- Let B > 0 be such that x = eB*is strongly plurisubharmonic on {x > o} ([5], Lemma 1). Theorem 3 in [5] is validwith obvious modifications on manifolds with corners. So we can approximate x m

C2 topology with \' C°° function satisfying:(a) x is an m - function on C(b) x is strongly plurisubharmonic on {~x > a}(c)x = X on x ({<*})(d)X > e B a o n {x> a}(e) to a critical value of x o r X\ac corresponds an unique critical point.

As C is compact the set of m— functions on C coinciding with x o n X~l{{a}) 1S

also open. Extend x to an open neighbourhood C (small enough) of C in X \ A suchthat the extension x' IS s ^ ^°° anc^ equals x o n X~1({a}) ^ C• As a is a regularvalue for x w e m a y write x' — eBx + h(x ~ Q) where h is C2. We approximate nowh\c in the C2 topology with a real analytic h* such that x* — e s^ + /i*(x — a) satisfies(a)-(e). By construction x* ' s r e a ' analytic.

To end the proof of (E\) (and of the assertion (A)) we use Theorem 3.1 on themanifold with boundary {R' < ift < R,<p > e} with x* a s rn~ function on x*~l{[a-,0\)(/? big enough). We only have to note that the index of a critical point of X* 1S ^ n

and the index of a critical point of type / of x* o n {^ = R-i X ^ Q } a^ s o ^ n ( s e e [5],page 5).Proof of (B) Let (A', n) be a triangulation of X compatible with tj)~1({Rl,})'^uwhich liniarizes (p i.e.:(i) K is an (abstract) simplicial complex and IT : jA'j i—> X a subanalytic homeomor-phism.(ii)V(T (closed) simplex f o TT^ is liniar.(iii) {Trflntcrjjvejv is compatible with {Rv < V' < R-v+i) Vf.

The existence of (A', 7r) follows from Shiota's theorem [12] (in fact a slight modi-fication of the argument on page 674-675). Now {ip o n = 0} is a polyhedron in |A'|.Refining the triangulation of |A'J (and to ease notation) we may suppose we are inthe following situation:

X underlines a simplicial complex K such that A, {4>~1({Ris})i {Rv < 4> 5: Ru+i}underlines simplicial subcomplexes L respectively K* respectively Kv such that L isfull in K and Lv := L ft Ku, Lv := LCi K* are full in Kv respectively in K* for any u.

Let W = N{L, A*') (where K' is a derived of K near L compatible with Ku, K* V//)be a simplicial neighbourhood of L in K (see [11], page 32). A simplex a E W iswritten uniquely as a join a = 0j of a j3 € A and a 7 £ N(L,K'). Also H^ : =

= N(L, K[) and W* := WOK; = N{L, A';')- A simplicial retract r : W H+ Lis defined as follows: let a; G <r = ^7. Then there exists uniques y 6 /3 and 2 6 7 suchthat j ; £yz. Put r(j-) = y. By definition r is compatible with WP\KU and WC\K'V W.

Define A : H' x / ^ W by /j(a:,*) - (1 - t)x + tr(x). This is well definedfin the sense that h takes values in W) and subanalytic. Moreover h(Wv x /) CWu, h(W; x /) C W; for any u.

Fix £* £ <S" such that V̂ i C M7. Let e < e1. Since 97 is liniar on simplices<^{h[x,t)) = (1 - t)(p(x) < ip{x) therefore h(Ve x /) C K- In particular A is adeformation retract of Vs. Let P C K be a closed subanalytic subset. Define Q =/i(P x /) which is also closed subanalytic (since Q C\ {ip < R^j is subanalytic W ) .Since h(h(x,u),t) = h(x,t + u(l - <)) we have h(Q x /) C Q therefore Q C\ A \s a,deformation retract of A so A is a deformation retract of A U Q and finally A U Q isa deformation retract of K.Proof of the affine algebraic case We choose now S to be a complex algebraicstratification and %jj to be a semialgebraic function (the rest of the definitions andnotations introduced throughout the Section remaining the same). Let P C X becompact subanalytic. Then there exists i/0 such that P C {0 < -R^} and 0 has nocritical values with respect to J7 on [/?yo,oo). Since {0 < R^} U A is a, deformationretract of X (see [4], page 132) it is enough to show that ({0 < #^},0) has n- trailproperty. Now using Lemma 2.6 (with D = 0) it is enough to show that for anyv < u0 denoting Ru , Ru+l with R', R the following is true:( f W - H I f l ' , / * ] ) , ^ - 1 ^ ' } ) has n - t r a i l property.In turn (f1) splits into two parts (bellow a = av and x = Xt>)'•

(Fi) ({4> < R,x > o}, {0 < R,X — Q}) has exact n— trail property with respect to{V1 = /?, x > ct} and

< 0 < /?, x ^ °}i 0~1({^}) has exact trail property with respect to {ifi <

{Fi) is exactly (Ej) above while (F2) is a consequence of the liniarization theoremof Shiota for \ (it is easy to see that x is a subanalytic function). I

Remark 4.1 The reason for which the telescopic neighbourhoods Ve are needed in theproof of the general case is that by Shiota's linearization theorem we can't concludethat:

{{Rv <• V' ^ Ru+i,Xv ^ °v}: ip~l([Rv, Rv+\])r\ A) has exact trail property with respectto {Rv < 0 < Rv+i, x» = ov} U (V- = Ru+i ,Xu<a,}

Remark 4.2 We proved that (X, A) has n— trail property whenever A C X and A 3Sing (A'). But this is true for general A as it follows by induction (see the beginningof the proof of the main theorem in [4]) and the obvious remark that (A U B, B) hasn- trail property if(B,Af\B) has it. In particular this gives:

Corollary 4.3 / / X is a Stein space A C X is analytic and if KA C A is closedsubanalytic. such that dim KA < dime A and KA is a deformation retract of A thereexists h'x C X closed subanalytic such that dim A'A- < dime X, Kx IS a deformationretract of X and moreover h'x H A D K^.

m

Acknowledgments

One of the authors (N.M.) would like to thank Professor Abdus Salam, the Interna-tional Atomic Energy Agency and UNESCO for hospitality at the International Centrefor Theoretical Physics, Trieste.

References

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[12] Shiota,M.: Piecewise linearization of real-valued subanalytic functions, Trans.A.M.S.,312(2) (1989), 663-679

[13] Shiota,M.: Piecewise linearization of subanalytic functions II, in Lecture Notesin Mathematics, vol. 1420 (1990), 247^307

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[15] Spanier.E.: Algebraic Topology, McGraw-Hill, 1966.

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