Algebraically Constructible Functions: Real Algebra and Topology

Algebraically constructible functions connect real algebra with the topology of algebraic sets. In this survey we present some history, definitions, properties, and algebraic characterizations of algebraically constructible functions, and a description of local obstructions for a topological space to be homeomorphic to a real algebraic set. More than three decades ago Sullivan proved that the link of every point in a real algebraic set has even Euler characteristic. Related topological invariants of real algebraic singularities have been defined by Akbulut and King using resolution towers and by Coste and Kurdyka using the real spectrum and stratifications. Sullivan’s discovery was motivated by a combinatorial formula for Stiefel-Whitney classes. Deligne interpreted these classes as natural transformations from constructible functions to homology. Constructible functions have interesting operations inherited from sheaf theory: sum, product, pullback, pushforward, duality, and integral. Duality is closely related to a topological link operator. To study the topology of algebraic sets the authors introduced algebraically constructible functions. Using the link operator we have defined many local invariants which generalize those of Akbulut-King and Coste-Kurdyka. Algebraically constructible functions are interesting from a purely algebraic viewpoint. From the theory of basic algebraic sets it follows that if a constructible function φ on an algebraic set of dimension d is divisible by 2 then φ is algebraically constructible. Parusiński and Szafraniec showed that algebraically constructible functions are precisely those constructible functions which are sums of signs of polynomials. Bonnard has given a characterization of algebraically constructible functions using fans, and she has investigated the number of polynomials necessary to represent an algebraically constructible function as a sum of signs of polynomials. Pennaneac’h has developed a theory of algebraically constructible chains using the real spectrum. In section 1 we briefly discuss the results of Sullivan, Akbulut-King, and Coste-Kurdyda. In the next section we define algebraically constructible functions and their operations. In section 3 we discuss the relations of algebraically constructible functions with real algebra. In the following section we describe how to generate our local topological invariants. In the final section we raise some questions for future research. Throughout we consider only algebraic subsets of affine space. This paper was originally written for the 2001 meeting in Rennes of the Real Algebraic and Analytic Geometry Network (RAAG), and it was posted on the RAAG website as a 1991 Mathematics Subject Classification. Primary: 14P25. Secondary: 14B05, 14P10. Research supported by CNRS and NSF grant DMS-9972094. 1 2 CLINT MCCRORY AND ADAM PARUSIŃSKI state-of-the-art survey. Related survey articles have been written recently by Coste [14] [15], Bonnard [8], and McCrory [26]. We thank Michel Coste for his encouragement and insight. 1. Akbulut-King Numbers Let X be a real semialgebraic set in R, and let x ∈ X. Let S(x, ε) be the sphere of radius ε > 0 in R centered at x. By the local conic structure lemma [6] (9.3.6), for ε sufficiently small the topological type of the space S(x, ε) ∩X is independent of ε. This space is called the link of x in X, and it is denoted by lk(x,X). Our starting point is Sullivan’s theorem [35]: Theorem 1.1. If X is a real algebraic set in R and x ∈ X then the Euler characteristic χ(lk(x,X)) is even. For example, the “theta space” X ⊂ R2, X = {(x, y) | x + y = 1} ∪ {(x, y) | − 1 ≤ x ≤ 1, y = 0}, is not homeomorphic to an algebraic set, for the link of the point (1, 0) (or the point (−1, 0)) in X is three points, which has odd Euler characteristic. Many proofs of Sullivan’s theorem have been published; see [12], [20], [5] (3.10.4), [6] (11.2.2), [18] (4.4). Sullivan’s original idea was to use complexification. First he proved that the link of x in the complexification XC has Euler characteristic 0, and then he used that lk(x,X) is the fixed point set of complex conjugation on lk(x,XC) to deduce that χ(lk(x,X)) ≡ χ(lk(x,XC)) (mod 2). Mather [25] (p. 221) gave a proof that the link L of a point in a complex algebraic set has Euler characteristic 0 by constructing a tangent vector field on L which integrates to a nontrivial flow of L. The following result puts Sullivan’s theorem in a more general context (cf. [3] (2.3.2)). Theorem 1.2. If X and Y are real algebraic sets with Y irreducible and f : X → Y is a regular map, there is an algebraic subset Z of Y with dimZ < dimY such that the Euler characteristic χ(f−1(y)) is constant mod 2 for y ∈ Y \ Z. In other words, the Euler characteristic is generically constant mod 2 in every family of real algebraic sets. To deduce Sullivan’s theorem as a corollary let Y = R, x0 ∈ X, and f(x) = (x − x0) 2. For y < 0 the fiber f−1(y) is empty, and for y > 0 sufficiently small, the fiber f−1(y) is lk(x0,X). Benedetti-Dedò [4] and Akbulut-King [2] proved that Sullivan’s condition is not only necessary but also sufficent in low dimensions: If X is a compact triangulable space of dimension less than or equal to 2, and the link of every point has even Euler characteristic, then X is homeomorphic to a real algebraic set. (The link of a point in a triangulable space is the boundary of a simplicial neighborhood.) A triangulable space such that the link of every point has even Euler characteristic is called an Euler space. Akbulut and King [3] showed that the situation in dimension 3 is more complicated. They defined four non-trivial topological invariants of a compact Euler space Y of dimension at ALGEBRAICALLY CONSTRUCTIBLE FUNCTIONS 3 most 2, ai(Y ) ∈ Z/2, i = 0, 1, 2, 3 (with ai(Y ) = 0 when dimY < 2). Let χ2(Y ) be the Euler characteristic mod 2. It is easy to see that if X is an Euler space then the link of every point of X is an Euler space. Theorem 1.3. A compact 3-dimensional triangulable topological space X is homeomorphic to a real algebraic set if and only if, for all x ∈ X, χ2(lk(x,X)) = 0 and ai(lk(x,X)) = 0, i = 0, 1, 2, 3. Akbulut and King’s invariants arise from a combinatorial analysis of the resolution of singularities of an algebraic set. The elementary definition of these Akbulut-King numbers and computations of examples can be found in Akbulut and King’s monograph [3], chapter VII, pages 190–197. (In the terminology of [3] (7.1.1), ai(lk(x,X)) is the mod 2 Euler characteristic of the link of x in the characteristic subspace Zi(X).) The depth of the method of resolution towers is shown by the remarkable result that the vanishing of the Akbulut-King numbers gives a sufficient condition for a triangulable 3-dimensional space to be homeomorphic to an algebraic set. Chapter I of [3] is an introduction to their methods, with informative examples. Another descendant of Sullivan’s theorem is due to Coste and Kurdyka [16]: Theorem 1.4. Let X be an algebraic set and let V be an irreducible algebraic subset. For x ∈ V the Euler characteristic of the link of x in X is generically constant mod 4: There is an algebraic subset W of V with dimW < dimV such that χ(lk(x,X)) is constant mod 4 for x ∈ V \W . This theorem was first proved by Coste [13] when dimX − dimV ≤ 2 using chains of specializations of points in the real spectrum. The general case was proved topologically using stratifying families of polynomials. It can also be proved using Akbulut and King’s topological resolution towers ([3], exercise on p. 192). Using the same techniques Coste and Kurdyka defined invariants mod 2 associated to chains X1 ⊂ X2 ⊂ · · · ⊂ Xk of algebraic subsets of X ([16], Theorem 4). Furthermore they used their mod 4 and mod 8 invariants to recover the Akbulut-King numbers. Using a relation between complex conjugation and the monodromy of the complex Milnor fibre of an ordered family of functions, the authors [27] reinterpreted and generalized the Coste-Kurdyka invariants as Euler characteristics of iterated links. 2. Constructible Functions Algebraically constructible functions were introduced by the authors [28] as a vehicle for using the ideas of Coste and Kurdyka to generate new Akbulut-King numbers. Let X be a real semialgebraic set. A constructible function on X is an integer-valued function φ : X → Z which can be written as a finite sum