Globalization and compactness of McCrory Parusiński conditions. Riccardo Ghiloni 1

Globalization and compactness of McCrory Parusiński conditions Riccardo Ghiloni 1 Department of Mathematics, University of Trento, 38050 Povo, Italy ghiloni@science.unitn.it Abstract Let X R n be a closed semialgebraic set and let fp(x) be the ring obtained from the characteristic function of X by the operations +,,, the half link operator and by the polynomial operations with rational coefficients which preserve finite formal sum of signs. McCrory and Parusiński proved that necessary conditions for X to be semialgebraically homeomorphic to a real algebraic set is that X is P euler, i.e, all the functions in fp(x) are integer valued. These conditions are local. In this paper, we give global versions of these conditions. For example, we show that, if X is P euler and M is a compact Nash submanifold of R n transverse to some semialgebraic Whitney stratification of X, then X M is P euler and, for each ϕ fp(x M), the Euler integral of ϕ is even. Moreover, we have the following result: Let F be a family of closed semialgebraic sets and let F c be the family of all X F such that X is compact. Suppose F is closed under the inverse images of regular maps. Then all the elements of F are P euler if and only if all the elements of F c are. Since it is known that every compact arc symmetric semialgebraic set is P euler, we infer that all the arc symmetric semialgebraic sets are P euler, answering affirmatively to a question by Kurdyka, McCrory and Parusiński. 1 Introduction In [10, 11], McCrory and Parusiński introduce and study the ring of algebraically constructible functions on a real algebraic set. The stability of this ring under the half link operator and some arithmetic operations with rational coefficients is used to define local topological invariants, the vanishing of which gives necessary conditions for a triangulable topological space to be homeomorphic to a real algebraic set. In this paper, we investigate some properties of these invariants. In order to place our results in the correct setting, we need to recall the method of McCrory and Parusiński. We will use the language of semialgebraic geometry, but one could work as well in the subanalytic setting (see Remark 3.4 at the end of the paper). Let X R n be a semialgebraic set (which we always consider locally closed). A function ϕ : X Q is constructible (over Q) if it has a presentation as a finite sum i m i1 Xi where, for each i, m i Q, X i is a closed semialgebraic subset of X and 1 Xi is the characteristic function of X i in X. Indicate by F(X, Q) the ring of such functions, with the usual addition and multiplication and by 1 X the identity of F(X, Q). Let ϕ = i m i1 Xi be an element of F(X, Q). Given a compact semialgebraic subset L of X, it is well defined the Euler integral of ϕ on L by L ϕ := i m iχ(x i L) where χ is the Euler characteristic. The link of ϕ at a point x X is defined by Λϕ(x) := lk(x,x) ϕ where lk(x, X) is the link of x in X obtained as intersection of X with a sufficiently small sphere of R n centered at x. It is easy to see that Λϕ F(X, Q). Define the half link operator Λ : F(X, Q) F(X, Q) by Λ := 1 2 Λ and denote by Λ(X) the subring of F(X, Q) obtained from 1 X by the operations +,, and Λ. Let X be a semialgebraic 1 The author is a member of GNSAGA of CNR, partially supported by MURST and European Research Training Network RAAG 2002 2006 (HPRN CT 00271). 1

set and let h : X X be a semialgebraic homeomorphism. For each ϕ F(X, Q), it holds ( Λϕ) h = Λ(ϕ h) so the pullback map h : F(X, Q) F(X, Q) induces an isomorphism between Λ(X ) and Λ(X). Suppose now X is a real algebraic set. A function ψ : X Z is called algebraically constructible if there is a finite family of proper regular maps f i : Z i X from real algebraic sets Z i to X and integers n i such that ψ can be written as follows: (1) ψ(x) = i n iχ(f 1 i (x)) for each x X. These functions form a subring of F(X, Q), denoted by A(X ). Remark that A(X ) Z X. The key property of A(X ) is the following consequence of resolution of singularities: Λ(A(X )) A(X ). Since 1 X A(X ), it follows that Λ(X ) A(X ) Z X and hence Λ(X) = h ( Λ(X )) Z X, i.e., all the functions in Λ(X) are integer valued. We call semialgebraic sets with this property as Λ euler (McCrory and Parusiński call them completely euler). In this way, a necessary condition for X to be semialgebraically homeomorphic to a real algebraic set is that X is Λ euler. The above construction can be improved. Let P be the set of polynomials in Q[t] which preserves finite formal sum of signs (see [11, Thm 4.2] for a complete description of P) and, for each P P, let P : F(X, Q) F(X, Q) be the map sending ϕ into P ϕ. Denote by P(X) the subring of F(X, Q) obtained from 1 X by the operations +,,, Λ and {P P P}. Observe that Λ(X) P(X). The semialgebraic set X is called P euler if P(X) Z X. One can prove that a function ϕ on the real algebraic set X belongs to A(X ) if and only if ϕ is a finite sum of signs of polynomials on X [5, 14]. This characterization of A(X ) implies that P (A(X )) A(X ) for each P P. It follows that P(X ) A(X ) Z X and hence P(X) = h ( P(X )) Z X. Thus, if X is semialgebraically homeomorphic to a real algebraic set, then X is not only Λ euler but also P euler. The conditions that X is Λ euler or P euler are local. The link operator has the following slice property (see [11, Section 1.3(d)]): (2) Let L be a closed semialgebraic subset of X. Suppose that a semialgebraic neighborhood of L in X is semialgebraically homeomorphic to the product of L with an interval. Then, for each ϕ P(X), Λϕ L = 2ϕ L Λ(ϕ L ). Applying (2) to the links of points of X, we obtain the following localization of the Λ euler and P euler conditions on X (see [11, Sections 2.2, 4.2]): (3) X is Λ euler (P euler resp.) if and only if, for each x X, the link L of x in X is Λ euler (P euler resp.) and, for each function ϕ in Λ(L) ( P(L) resp.), ϕ 0 (mod 2). L One can prove that a finite list of these Z/2 obstructions is necessary and sufficient for X to be Λ euler (P euler resp.). In dimension 3, these Z/2 obstructions recover exactly the Akbulut King numerical conditions so X is semialgebraically homeomorphic to a real algebraic set if and only if it is Λ euler [1, 10]. In particular, every Λ euler semialgebraic set of dimension 3 is P euler also. In higher dimension, the P euler condition is strictly stronger than the Λ euler condition. In fact, the number of independent Z/2 obstructions for a 4 dimensional semialgebraic set to be Λ euler is 2 29 29, while such a number increases to 2 43 43 in the P euler case. The method of McCrory and Parusiński can be used to study the local topological properties of geometric objects more general than real algebraic sets as arc symmetric semialgebraic sets introduced by Kurdyka [7, 8] and real analytic sets [13]. Suppose X is an arc symmetric semialgebraic set. A function ψ : X Z is Nash constructible if it has a presentation as in (1) where, for each i, Z i is a connected component of a real 2

algebraic set. Arc symmetric semialgebraic sets admit resolution of singularities so 1 X is Nash constructible and the ring of Nash constructible functions on X is preserved by Λ. In particular, X is Λ euler. A similar argument can be repeated and the same conclusion holds if X is a real analytic set. It follows that all the arc symmetric semialgebraic sets and all the real analytic sets are Λ euler. In dimension 3, these sets are P euler also. In the compact arc symmetric case, one can say some more. In [3], Bonnard shows that a function on a compact arc symmetric semialgebraic set is Nash constructible if and only if it is a finite sum of signs of blow Nash functions. It follows that all the compact arc symmetric semialgebraic sets are P euler also. This paper deals with the following two problems: PROBLEM A ([9, p. 21], [12, p. 12]). Is a noncompact arc symmetric semialgebraic set of dimension 4 P euler? Is a real analytic set of dimension 4 P euler? PROBLEM B. Is it possible to infer obstructions of global nature on the topology of real algebraic, arc symmetric semialgebraic, or real analytic sets from the McCrory Parusiński local conditions? Taking property (2) as starting point, we introduce a class of subsets of X, called boundary slices of X. The nature of these subsets of X is global and they include the iterated links of points of X. By a simple generalization of (2) and a Stokes type theorem for the link operator (see section 3), we prove the following globalization theorem: if X is Λ euler (P euler resp.), i.e, the McCrory Parusiński invariants corresponding to the Λ euler ( P euler resp.) condition vanish on the links of points of X, then the same invariants vanish on the boundary slices of X. This generalizes (3). Let F be a family of semialgebraic sets closed under the inverse images of regular maps and let F c be the family of all X F such that X is compact. The preceding theorem implies the following result: if all the elements of F c are P euler, then all the elements of F are P euler also. This result applies to the family of all arc symmetric semialgebraic sets and to the family of all real analytic sets. It follows that the first part of PROBLEM A has a positive answer. Moreover, in order to prove that all the real analytic sets are P euler, it suffices to show that all the compact real analytic sets described by global equations are P euler. Using the globalization theorem, we deduce also obstructions of global nature on the topology of Λ euler (P euler resp.) semialgebraic sets (see Remark 2.12). The mentioned results are presented in the next section. The proofs are given in section 3. 2 Globalization theorem and its applications In what follows, we use the symbol Θ to indicate either Λ or P. Let us introduce the notions of Θ boundary and of boundary slice. Let X R n be a semialgebraic set. Definition 2.1 We say that X is a Θ boundary if it is compact, Θ euler and, for each ϕ Θ(X), the Euler integral of ϕ is even. Let Y be a locally closed semialgebraic subset of X and let j be a nonnegative integer. We say that Y is a j slice of X if, for each y Y, there are an open semialgebraic neighborhood U of y in X, a semialgebraic set V and a semialgebraic homeomorphism h : V ( 1, 1) j U such that h(v {0}) = U Y. Such a homeomorphism is said to be a slicing chart for Y. Observe that a k slice of a j slice of X is a (k +j) slice of X. A slice of X is a j slice of X for some j. Let W be another semialgebraic subset of X. Denote by int(w ) and W the interior and the topological boundary of W in X respectively. We say that W is a regular domain of X if it is 3

compact, W is a 1 slice of X and, for each w W, there are an open semialgebraic neighborhood U of w in X and a slicing chart h : V ( 1, 1) U for W such that h(v ( 1, 0)) = U int(w ). Definition 2.2 Let L be a j slice of the semialgebraic set X for some j > 0. We say that L is a boundary slice of X if there exist a (j 1) slice Y of X containing L and a regular domain W of Y such that L is the topological boundary of W in Y. Remark 2.3 The local conic structure theorem implies that all the iterated links of points of X, viewed as intersections of X with small spheres of R n, are boundary slices of X. Moreover, if X is Θ euler, then (3) implies that such iterated links are Θ boundaries. The following theorem is the main result of this paper. Theorem 2.4 Every boundary slice of a Θ euler semialgebraic set is a Θ boundary. In the following two subsections, we present some applications of the preceding theorem. 2.1 Compactness Let F be a family of semialgebraic sets. Recall that F is called Θ euler if each element of F is Θ euler. We say that F is locally algebraically stable if, for each element X R n of F and for each x X, there is a neighborhood U of x in R n with the following property: for each regular map f : R n R n whose image is in U, f 1 (X) is again an element of F. Theorem 2.5 Let F be a locally algebraically stable family of semialgebraic sets and let F c be the family of all X F such that X is compact. Then F is Θ euler if and only if F c is Θ euler. Applying this result to the family of all arc symmetric semialgebraic sets and using [3, Thm 2], we obtain: Corollary 2.6 Every arc symmetric semialgebraic set is P euler. A real analytic set V R n is called C analytic if it has a global equation, i.e., if there exist an open subset Ω of R n containing V and an analytic function F : Ω R such that V = F 1 (0) (see [13, p. 104] for the reason of such a nomenclature). Suppose the following holds: all the compact C analytic sets are P euler. By Theorem 2.5, every C analytic set would be P euler. On the other hand, a real analytic set is locally C analytic so the same would be true for every real analytic set. We have just proved the following result. Corollary 2.7 Let An be the family of all real analytic sets and let C An c be the family of all compact C analytic sets. Then An is P euler if and only if C An c is P euler. Let V be a compact C analytic set. We say that a function ψ : V Z is C analytically constructible if it has a presentation as in (1) where, for each i, Z i is a compact C analytic set and f i is a real analytic map. Question 2.8 Does there exist for each V C An c a class C V of R valued functions on V such that a function ψ : V Z is C analytically constructible if and only if ψ is a finite sum of signs of elements of C V? Observe that a positive answer to this question and Corollary 2.7 would imply that all the real analytic sets are P euler. 4

2.2 Topological obstructions of global nature We will use standard notions from real Nash geometry (see [2]). By Nash manifold, we mean a Nash submanifold of some R n. A Nash manifold and a Nash map between Nash manifolds can be regarded in the natural way as a smooth manifold and a smooth map respectively. Let N be a smooth manifold, let X be a subset of N and let f : M N be a smooth map from a compact smooth manifold M to N. We say that f can be moved away from X if it is unoriented bordant to a smooth map whose image is disjoint from X (see [4] for the definition of unoriented bordism of smooth maps). Theorem 2.9 Let N be a Nash manifold, let X be a closed semialgebraic subset of N of codimension r > 0 and let f : M N be a Nash map from a compact Nash manifold M of dimension m r to N transverse to some semialgebraic Whitney stratification of X in N. Suppose that X is Θ euler and one of the following two conditions holds: (i) H k (N, Z/2) = {0} for each k {r, r + 1,..., m}, (ii) f can be moved away from X. Then f 1 (X) is a Θ boundary. Let T be a topological space. Recall that T is euler if, for each p T, χ(t, T \ {p}) is odd. If T is a semialgebraic set, then this property is equivalent to say that Λ1 T is integer valued (Sullivan s condition). We have: Corollary 2.10 Let X be a Θ euler closed semialgebraic subset of R n of dimension < n and let M be a compact smooth submanifold of R n, which intersects transversally a Whitney stratification X of X in R n. The following holds: (i) X M is euler and has even Euler characteristic. (ii) If M is a compact Nash submanifold of R n and all the strata of X are semialgebraic, then X M is a Θ boundary. Remark 2.11 In order to obtain (i) in the preceding corollary, it suffices that X is euler. Remark 2.12 The preceding results give obstructions for a semialgebraic set X to be semialgebraically homeomorphic to a real algebraic set starting from informations of global nature. However, it is important to remark that such obstructions are induced from the McCrory Parusiński local conditions. In particular, they cannot be used to construct examples (if they exist) of semialgebraic sets locally, but not globally, semialgebraically homeomorphic to real algebraic sets (see [12, p. 12, Question 2]). 3 Proofs We need some preparations. Given a semialgebraic set X, we define the operator Ω j : F(X, Q) F(X, Q) by Ω j (ϕ) := ϕ Λϕ if j is odd and by Ω j (ϕ) := Λϕ if j is even. Lemma 3.1 Let X be a semialgebraic set and let Y be a j slice of X. The following holds: (i) Let y Y, let U be an open semialgebraic neighborhood of y in X and let h : V ( 1, 1) j U be a slicing chart for Y. Denote by h : V ( 1, 1) j X the composition of h with the inclusion map U X and by π : V ( 1, 1) j V the natural projection. Then, for each Φ Θ(X), there exists ϕ Θ(V ) such that Φ h = ϕ π. (ii) Every function in Θ(Y ) extends to a function in Θ(X). In particular, if X is Θ euler, then Y is Θ euler also. 5

Proof. We give the proof only when Θ is P. The case Θ = Λ is similar. Let P 0 (X) P 1 (X) P 2 (X)... be the exaustive sequence of subrings of P(X) defined as follows: P0 (X) is the ring generated by 1 X and, for each nonnegative integer k, Pk+1 (X) is the smallest ring containing P k (X) { Λϕ ϕ P k (X)} and closed under the operations {P P P}. Observe that 1 X h = 1 V π. Let Φ P k (X). Suppose there is ϕ P k (V ) such that Φ h = ϕ π. For each point (v, t) V ( 1, 1) j, the link of (v, t) in V ( 1, 1) j is the j iterated suspension of the link of v in V so Λ(ϕ π)(v, t) = Ω j (ϕ)(v). Since ( ΛΦ) h = Λ(Φ h ), it follows that ( ΛΦ) h = Ω j (ϕ) π, where Ω j (ϕ) P k+1 (V ). Using this fact inductively on k, we obtain (i). Let us show, by induction on k, that every function in P k (Y ) extends to a function in P k (X). The case k = 0 is evident: 1 Y = 1 X Y. Let ϕ P k (Y ) and let Φ P k (X) be an extension of ϕ. We must only prove that Λϕ admits an extension in P k+1 (X). Let y Y, let h : V ( 1, 1) j U and h : V ( 1, 1) j X be as in (i) and let y := h 1 (y). Observe that Λϕ(y) = (( Λϕ) h V {0} )(y ) = Λ(ϕ h V {0} )(y ) and, by (i), Λ(ϕ h V {0} )(y ) = Ω j (Φ h )(y ) so Λϕ(y) = Ω j (Φ h )(y ) = Ω j (Φ)(y). The function Ω j (Φ) is the desired extension of Λϕ. The next result is a Stokes type theorem for the link operator. Lemma 3.2 Let X be a semialgebraic set, let W be a regular domain of X and let ϕ P(X). Then Λϕ = ϕ. W Proof. Let 1 W and 1 W be the characteristic functions of W and of W in X respectively. From Lemma 3.1,(i) with j = 1, it follows that Λ(ϕ 1 W ) = (Λϕ) 1 W (Λϕ) 1 W + ϕ 1 W and (Λϕ) 1 W = 2ϕ 1 W Λ(ϕ 1 W ). In particular, Λ(ϕ 1 W ) = (Λϕ) 1 W ϕ 1 W + Λ(ϕ 1 W ). Recall that the Euler integral of the link of a constructible function with compact support is zero (see Corollary 1.3,(iii) of [10]). We have: 0 = X Λ(ϕ 1 W ) = W Λϕ W ϕ + X Λ(ϕ 1 W ) = W Λϕ W ϕ. W We are now in position to prove our theorems. Proof of Theorem 2.4. Let X be a Θ euler semialgebraic set, let Y be a slice of X and let W be a regular domain of Y. We must prove that the topological boundary W of W in Y is a Θ boundary. By Lemma 3.1,(ii), we know that Y and W are Θ euler and, for each ϕ P( W ), there is Φ P(Y ) such that Φ W = ϕ. Fix ϕ P( W ) and an extension Φ P(Y ) of ϕ. Lemma 3.2 implies that W ϕ = ΛΦ. Since all W the values of ΛΦ are even, we infer that ϕ is even also. W Proof of Theorem 2.5. Suppose F c is Θ euler. Let X R n be an element of F \F c. By (3), we must prove that, for each x X, the link lk(x, X) of x in X is a Θ boundary. Fix x X. If the local dimension dim x (X) is n, then lk(x, X) is the standard sphere S n 1, which is a Θ boundary. Suppose dim x (X) < n. Let U be an open neighborhood of x in R n such that, if f : R n R n is a regular map with f(r n ) U, then f 1 (X) is an element of F. Identify R n with R n {0} R n R = R n+1 and, for each positive real number ε, denote by S n 1 (x, ε) (resp. S n (x, ε)) the ε sphere of R n (resp. R n+1 ) centered in x. Choose a positive real number ε and a semialgebraic Whitney stratification X of X such that the closed ε ball of R n centered at x is contained in U, S n 1 (x, ε) X is equal to lk(x, X) and S n 1 (x, ε) is transverse to X in R n. Extend the inclusion map S n 1 (x, ε) R n to a smooth map g : S n (x, ε) R n in such a way 6

that its image is contained in U. By the Thom transversality theorem and standard algebraic approximation results, we may suppose that g is a regular map transverse to X in R n. Let p S n 1 (x, ε) \ X and let ψ : R n S n (x, ε) \ {p} be a biregular isomorphism such that ψ(r n 1 ) = S n 1 (x, ε) \ {p} where R n 1 := {x R n x n = 0}. Define the regular map f : R n R n by f := g ψ. Since the image of f is contained in U and f 1 (X) is compact, we have that f 1 (X) is an element of F c and hence it is Θ euler. Let W := f 1 (X) {x R n x n 0}. Since f R n 1 is transverse to X in R n, the semialgebraic version of Thom s first isotopy lemma [6] ensures that the topological boundary W of W in f 1 (X) is a boundary slice of f 1 (X) and coincides with f 1 (X) R n 1 = ψ 1 (lk(x, X)). In particular, W is semialgebraically homeomorphic to lk(x, X). By Theorem 2.4, W (and hence lk(x, X)) is a Θ boundary. Proof of Theorem 2.9. First, we show that (i) implies (ii). Suppose (i) holds. By [4, Thm 17.1] and condition (i), there are a finite family {V i } k i=1 of compact smooth manifolds and a finite family {g i : W i N} k i=1 of smooth maps from compact smooth manifolds W i to N such that, for each i {1,..., k}, dim(w i ) < r, dim(v i ) + dim(w i ) = m and, denoting by π i : V i W i W i the natural projections, f is unoriented bordant to the disjoint union map g : k i=1 (V i W i ) N of g 1 π 1, g 2 π 2,..., g k π k. Let X be a semialgebraic Whitney stratification of X such that f is transverse to X in N. By the Thom transversality theorem, we may suppose that, for each i {1,..., k}, g i is transverse to X in N, i.e, the image of g i is disjoint from X. It follows that the image of g is also disjoint from X and hence (i) holds. Let us prove the lemma assuming that (ii) is verified. Thanks to this condition, there are a compact smooth manifold P with boundary and a smooth map F : P N such that P is the disjoint union of M and a compact smooth manifold M, F M = f and F (M ) X =. Double the bordism F obtaining a compact smooth manifold P without boundary and a smooth map F : P N. Consider P as a subset of P in such a way that F P = F. Applying the Thom transversality theorem and standard Nash approximation results, we may suppose that P is a compact Nash manifold, P is a semialgebraic subset of P and F is a Nash map transverse to X in N. Let us show that (F ) 1 (X) is Θ euler. Let Γ F be the graph of F and let P X be the semialgebraic Whitney stratification of P X defined by P X := {P S} S X. Observe that (F ) 1 (X) is semialgebraically homeomorphic to Γ F (P X) and Γ F is transverse to P X in P N. The latter fact and the semialgebraic version of Thom s first isotopy lemma implies that Γ F (P X) is a slice of P X. From the first part of Lemma 3.1, it follows that P X is Θ euler and, from the second part of the same lemma, we infer that Γ F (P X) (and hence (F ) 1 (X)) is Θ euler as desired. Define W := P (F ) 1 (X). The intersection W M is empty so the topological boundary W of W in (F ) 1 (X) is contained in W M = f 1 (X). The map f is transverse to X in N so, using again the semialgebraic version of Thom s first isotopy lemma, we have that W = f 1 (X) and W is a regular domain of (F ) 1 (X). In particular, f 1 (X) is a boundary slice of (F ) 1 (X). By Theorem 2.4, it follows that f 1 (X) is a Θ boundary. Remark 3.3 If, in Theorem 2.9, X is euler instead of Θ euler, then we can conclude that f 1 (X) is euler and has even Euler characteristic. This follows from an easy review of the preceding proofs. Proof of Corollary 2.10. Assertion (ii) is an easy consequence of Theorem 2.9. Let us prove (i) assuming that X is euler. Let X be a semialgebraic Whitney stratification of X. Thanks to the Thom transversality theorem and Thom s first isotopy lemma, there exists a compact smooth manifold M of R n arbitrarily close to M such that M is transverse to X in R n and X M is homeomorphic to X M. By standard Nash 7

approximation results, we may also suppose that M is a compact Nash submanifold of R n. The preceding remark implies that X M (and hence X M) is euler and has even Euler characteristic. Remark 3.4 Except for Corollary 2.6, in sections 2 and 3, the words semialgebraic and Nash can be always replaced by subanalytic and real analytic respectively. Acknowledgements We wish to express our gratitude to Michel Coste for several useful discussions. References [1] S. Akbulut, H.C. King, Topology of Real Algebraic Sets, Mathematical Sciences Research Institute Publications, no. 25, Springer Verlag, New York, 1992. [2] J. Bochnak, M. Coste, M.-F. Roy, Real algebraic geometry. Translated from the 1987 French original. Revised by the authors, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 36. Springer-Verlag, Berlin, 1998. [3] I. Bonnard, Nash constructible functions, Manuscripta Math. 112 (2003), no. 1, 55 75. [4] R.J. Conner, E.E. Floyd, Differential periodic maps, Ergeb. Math. Grenzgeb. (2) 33, Springer Verlag, Berlin 1964. [5] M. Coste, K. Kurdyka, Le discriminant d un morphisme de variétés algébriques réelles. (French) [The discriminant of a morphism of real algebraic varieties] Topology 37 (1998), no. 2, 393 399. [6] M. Coste, M. Shiota, Thom s first isotopy lemma: a semialgebraic version, with uniform bound, Real analytic and algebraic geometry (Trento, 1992), 83 101, de Gruyter, Berlin, 1995. [7] K. Kurdyka, Ensembles semi-algébriques symétriques par arcs. (French) [Arcwise symmetric semi-algebraic sets] Math. Ann. 282 (1988), no. 3, 445 462. [8] K. Kurdyka, Injective endomorphisms of real algebraic sets are surjective. Math. Ann. 313 (1999), no. 1, 69 82. [9] K. Kurdyka, A. Parusiński, Arc symmetric sets and arc analytic mappings, (to appear) [10] C. McCrory, A. Parusiński, Algebraically constructible functions, Ann. Sci. Ec. Norm. Sup. 30 (1997), 527 552. [11] C. McCrory, A. Parusiński, Topology of real algebraic sets of dimension 4: necessary conditions, Topology 39 (2000), no. 3, 495 523. [12] C. McCrory, A. Parusiński, Algebraically Constructible Functions: Real Algebra and Topology, Real Algebraic and Analytic Geometry RAAG 01, Rennes, june 11 15th, 2001, electronic surveys, http://www.math.univ-rennes1.fr/geomreel/raag01/surveys/ mccpar.pdf [13] R. Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Mathematics, No. 25, Springer Verlag, Berlin New York, 1966. [14] A. Parusiński, Z. Szafraniec, Algebraically constructible functions and signs of polynomials, Manuscripta Math. 93 (1997), no. 4, 443 456. 8