Clint McCrory

Algebraically constructible functions

Abstract An algebraic version of Kashiwara and Schapiras calculus of constructible functions is used to describe local topological properties of real algebraic sets, including Akbulut and Kings numerical conditions for a stratified set of dimension three to be algebraic. These properties, which include generalizations of the invariants modulo 4, 8 and 16 of Coste and Kurdyka, are defined using the link operator on the ring of constructible functions.

STIEFEL-WHITNEY CLASSES AND THE CONORMAL CYCLE OF A SINGULAR VARIETY

A geometric construction of Sullivans Stiefel-Whitney homology classes of a real analytic variety X is given by means of the conormal cycle of an embedding of X in a smooth variety. We prove that the Stiefel-Whitney classes define additive natural transformations from certain constructible functions to homology. We also show that, for a complex analytic variety, these classes are the mod 2 reductions of the Chern-MacPherson classes. We present a new definition of the Stiefel-Whitney homology classes of a possibly singular real analytic variety X. The original definition, due to Sullivan [S], involves a triangulation of X; its geometric meaning is unclear. Our definition uses the conormal cycle of an embedding of X in a smooth variety. The conormal cycle of a subanalytic subset X of an analytic manifold M was defined by the first author [F4] using geometric measure theory. The conormal cycle is an integral current representing (up to sign) Kashiwaras characteristic cycle of the sheaf DMRX [F4, 4.7]. Our construction of the Stiefel-Whitney classes is based on the fundamental observation that the conormal cycle of a real analytic variety is antipodally symmetric mod 2. We use this observation to give a definition of Stiefel-Whitney classes which is parallel to the first authors definition of the Chern-Schwartz-MacPherson classes of a complex analytic variety [F5]. (This definition of Chern classes is related to earlier work of Brylinski-Dubson-Kashiwara [BDK] and Sabbah [Sa].) In fact we show that the Stiefel-Whitney homology classes of a complex analytic variety are the mod 2 reductions of the Chern classes. We prove that our Stiefel-Whitney classes satisfy axioms similar to the Deligne-Grothendieck axioms for Chern classes, and we prove a specialization formula for the Stiefel-Whitney classes of a family of varieties. We show that the Stiefel-Whitney classes of an affine real analytic variety X are represented by the polar cycles of X, introduced for simplicial spaces by Banchoff [B] and McCrory [Mc], and we give a new proof of the combinatorial formula for Stiefel-Whitney classes of manifolds. Our central result is a specialization formula for the conormal cycle (Theorem 3.7). We use this formula to prove the basic results of Kashiwara and Schapiras calculus of subanalytically constructible functions [Sc], [KS, 9.7], as well as the pushforward and specialization formulas for Stiefel-Whitney classes. Received by the editors October 2, 1995. 1991 Mathematics Subject Classification. Primary 14P25, 57R20; Secondary 14P15, 49Q15.

COUNTING TRITANGENT PLANES OF SPACE CURVES

LET C be a smooth simple closed curve in Iw3. A tritangent plane of C is a plane in W3 which is tangent to C at exactly three points. A stall x of C is a point of C at which the torsion of C is zero. We will say that a stall x is transoerse if the curvature of C is non-zero at x, the derivative of the torsion of C is non-zero at x, and the osculating plane P of C at x is transverse to C away from x. If x is a transverse stall of C then an interval of C about x lies on one side of the osculating plane P of C at x, so P intersects Cat an even number 2n of points other than x. The integer n = n(x, C) is the index of the transverse stall x of C. Let Coc(S’, rW3) be the space of C” maps ~1: S’ -+ W3 with the Whitney topology.

Complex monodromy and the topology of real algebraic sets

A relation between the Euler characteristics of the Milnorfibres of a real analytic function is derived from a simple identity involvingcomplex monodromy and complex conjugation. A corollary is the result of Costeand Kurdyka that the Euler characteristic of the local link of an irreduciblealgebraic subset of a real algebraic set is generically constant modulo 4. Asimilar relation for iterated Milnor fibres of ordered sets of functions isused to define topological invariants of ordered collections of algebraicsubsets.

Topology of real algebraic sets of dimension 4: necessary conditions

Abstract 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 −43 independent characteristic numbers mod 2, which generalize the Akbulut–King numbers in dimension three.

Cobordism operations and singularities of maps

If ƒ is a differentiable map of smooth manifolds, the critical set 2(f) is not a manifold, in general. However, there is a canonical resolution of the singularities of 2(f) (for generic ƒ), due to I. Porteous [6]. This resolution can be used to give a geometric description of T. torn Diecks Steenrod operations in unoriented cobordism [7]. This was suggested to me by Jack Morava, as a parallel to my discription of ordinary mod 2 Steenrod operations using branching cycles of maps of «-circuits [5] .

The Gauss map of a genus three theta divisor

A smooth complex curve is determined by the Gauss map of the theta divisor of the Jacobian variety of the curve. The Gauss map is invariant with respect to the (-1)-map of the Jacobian. We show that for a generic genus three curve the Gauss map is locally Z/2-stable. One method of proof is to analyze the first-order Z/2-deformations of the Gauss map of a hyperelliptic theta divisor

An axiomatic proof of Stiefel’s conjecture

Stiefels combinatorial formula for the Stiefel-Whitney homology classes of a smooth manifold is proved, by verifying that the classes defined by his formula satisfy axioms which characterize the Stiefel-Whitney classes.

Conic lagrangian singularities

Abstract We classify conic Lagrangian holomorphic map germs with Z 2 symmetry, in terms of homo geneous generating families. The motivating example is the Gauss map of the theta divisor of a principally polarized abelian variety.