1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

270 M. GRINBERG AND R. MACPHERSON, EULER CHARACTERISTICS

near the smaller stratum, which assure that the topology of S is locally constant

along each stratum. (In other words, that the group of stratification-preserving

homeomorphisms of Z acts transitively on each stratum.)

We define a real algebraic variety X as a subset of some Euclidean space !Rn,

cut out by polynomial equations. A subset S C !Rn is called semi-algebraic if it is

cut out by polynomial equations and inequalities.

Let X be a real algebraic manifold (i.e., a smooth real algebraic variety), and

Z C X a closed semi-algebraic subset. A semi-algebraic stratification of Z is a

stratification whose strata are semi-algebraic sets. One can show that any subdivi-

sion of Z into finitely many semi-algebraic sets can be refined to a semi-algebraic

stratification.

Exercise: The Whitney cusp. Let Z c JR^3 be the locus

{(x, y, z ) I y^2 = x^2 (z^2 - x)}

(see Figure 1). Show that any stratification of Z must have at least 7 strata.

z

Figure 1

1.3. Transversality

The transversality condition in Theorem 1.1 means (as usual) that if A is a stratum

of S1, Bis a stratum of S2, and xis in An B , then TxA + TxB = TxX.