Topology of Real Algebraic Sets of Dimension 4: Necessary Conditions

Operators on the ring of algebraically constructible functions are used to compute local obstructions for a four-dimensional semialgebraic set to be homeomorphic to a real algebraic set. The link operator and arithmetic operators yield 2 − 43 independent characteristic numbers mod 2, which generalize the Akbulut-King numbers in dimension three. The ring of algebraically constructible functions on a real algebraic set was introduced in [5]. The link operator on this ring was used to give a new description of the AkbulutKing numbers of three-dimensional stratified sets [2], as well as to generalize the topological conditions on algebraic sets discovered by Coste and Kurdyka [4]. Akbulut and Kurdyka have asked what invariants can be constructed by our method for four-dimensional sets. In this paper we produce a large number of independent new local topological conditions satisfied by algebraic sets of dimension four. Thus, in particular, there are four-dimensional semialgebraic sets which have vanishing Akbulut-King invariants, but which are not homeomorphic to algebraic sets. We do not know whether a four-dimensional semialgebraic set satisfying all of our conditions, as well as the Akbulut-King conditions, must be homeomorphic to an algebraic set. The properties of the link operator Λ on constructible functions are reviewed in section 1. The main result of [5] (see also [6], [7]) is that the operator Λ̃ = 1 2 Λ preserves the set of algebraically constructible functions: If φ is algebraically constructible, so is Λ̃φ. Thus, in particular, if X is a real algebraic set with characteristic function 1X , every function obtained from 1X using the arithmetic operations +, −, ∗, and the operator Λ̃ is integer-valued. Sets with this property we call completely euler. The property that Λ̃1X is integer-valued is equivalent to Sullivan’s condition [8] that X is euler: for all x ∈ X, the link of x in X has even euler characteristic. In section 2 we show how to construct systematically invariants of a set X which vanish if and only if X is completely euler. These invariants are local, and if we assume that all the links of points of X are completely euler, then X is completely euler if and only if a finite list of mod 2 characteristic numbers vanish for each link. For X of dimension at most 4 we find 229 − 29 such characteristic numbers. Section 3 contains a proof of the independence of these numbers, by construction of examples which distinguish them. Date: September 7, 1998. 1991 Mathematics Subject Classification. Primary: 14P25. Secondary: 14B05, 14P10. Research supported by CNRS. First author also supported by NSF grant DMS-9628522. 1 2 CLINT MCCRORY AND ADAM PARUSIŃSKI There are also arithmetic operators with rational coefficients which preserve the set of algebraically constructible functions. For example, if φ is algebraically constructible, then so is P (φ) = 1 2 (φ 4 −φ2). In section 4 we characterize such operators, and we show that P is the only such operator which gives new conditions on the topology of 4-dimensional algebraic sets. Using the operator P together with the operator Λ̃, we enhance our previous construction to produce 243 − 43 independent local characteristic numbers which vanish for algebraic sets of dimension ≤ 4. We work with real semialgebraic sets in euclidean space, semialgebraic Whitney stratifications, and semialgebraically constructible functions. We use the foundational results that the link of a point in a semialgebraic set is well-defined up to semialgebraic homeomorphism, and that a semialgebraic set has a semialgebraic, locally semialgebraically trivial Whitney stratification (cf. [4]). Since semialgebraic sets are triangulable and our constructions are purely topological, we could just as well work with piecewise-linear sets, stratifications, and constructible functions. It is not natural to assume that a real semialgebraic set has the same local dimension at every point, so we must be careful in dealing with dimension. By the dimension of a semialgebraic set X we mean its topological dimension, denoted dimX. If T is a semialgebraic triangulation of X, then dimX is the maximum dimension of a simplex of T . For x ∈ X, the local dimension dimxX of X at x is the maximum dimension of a closed simplex of T containing x. Thus dimX = max{dimxX, x ∈ X}, and if L is the link of x in X, then dimL = dimxX − 1, which may be less than dimX − 1. 1. Algebraically Constructible Functions 1.1. Definition and main properties. Let X ⊂ Rn be a real algebraic set. Following [5] we say that an integer-valued function φ : X → Z is algebraically constructible if there exists a finite collection of algebraic sets Zi and proper regular morphisms fi : Zi → X such that φ admits a presentation as a finite sum φ(x) = ∑ miχ(f −1 i (x)) (1.1) with integer coefficients mi, where χ is the euler characteristic. We recall from [6], [7] that φ : X → Z is algebraically constructible if and only if there exists a finite set of polynomials g1, . . . , gs ∈ R[x1, . . . , xn] such that φ(x) = sgn g1(x) + . . .+ sgn gs(x). (1.2) The set of algebraically constructible functions on X forms a ring, which we denote by A(X). An algebraically constructible function is constructible in the usual sense; that is, it is a finite combination of characteristic functions of semialgebraic subsets Xi of X, φ = ∑ ni1Xi (1.3) with integer coefficients ni. The converse is not true; there exist constructible functions which are not algebraically constructible (see [5]). Let X be a semialgebraic set, and let φ be a constructible function on X given by (1.3). Without loss of generality we may assume that all Xi are closed in X. If, moreover, all Xi REAL ALGEBRAIC SETS OF DIMENSION 4 3 are compact then we define the euler integral of φ by ∫ φdχ = ∑ niχ(Xi). It is easy to see that ∫ φdχ is well-defined; it depends only on φ and not on the presentation (1.3). The euler integral is uniquely determined by the properties that it is linear, ∫ φ+ ψ dχ = ∫ φdχ+ ∫ ψ dχ, and the euler integral of the characteristic function of a compact semialgebraic set is its euler characteristic, ∫ 1W dχ = χ(W ). If φ is given by (1.3), then for x ∈ X, the link of φ at x is defined by Λφ(x) = ∑ niχ(Xi ∩ S(x, ε)), where S(x, ε) is a sufficiently small sphere centered at x. Thus