LECTURE 2. 277Show that the covector dy is not generic at the origin, but that any covector not
proportional to dy is.The following exercise indicates the kind of pathology we try to avoid by re-
stricting our discussion to semi-algebraic stratifications.Exercise 2.3.2. Give an example of a (non semi-algebraic, Whitney) stratification
of IR^2 , such that the origin is a stratum, but there are no generic covectors at the
origin.We denote by Ai C As the set of all generic conormals to S, and by A~ the
union of the Ai over all SES.
Exercise 2.3.3. Let S be a semi-algebraic stratification of a real algebraic manifold
X. Show that A~ is the smooth part of As.2.4. The half-link
Let X be a real algebraic manifold, and S a semi-algebraic stratification of X. Fix
a stratum S, a point p E S, and a generic covector ~ E Ai at p. Associated to
~' there is a subset H(~) C X, called the half-link of~' which plays a key role in
stratified Morse theory. It is defined as an intersection of three things:
Here N denotes a normal slice to S at p, i.e., any submanifold of X of dimension
dim X - dim S, passing through p and transverse to S. The set B, is a closed ball
of radius E in X , centered around p (we need to fix a Riemannian metric on X for
this). Lastly, g-^1 (8) is a level surface of some smooth function g : X __,IR, such
that g(p) = 0 and dpg = ~· We assume that E and 8 satisfy 0 < 8 « E « l.
The definition of H ( ~) calls for a number of auxiliary choices (the normal slice,
the metric, the function g, and the numbers E and 8). However, for all practical
purposes, the result is independent of the choices. More precisely, the stratification
of X induces a stratification of H(~), and H(~) is well defined up to a stratification-
preserving homeomorphism.
Exercise. Show that H(~) = 0 when dimS = dimX.
2.5. The characteristic cycle
We can now define the characteristic cycle of an algebraically constructible function.
Let (X,S) b e as above. Assume, in addition, that Xis oriented. Set d = dimX.
Let f : X __, Z be a function constructible with respect to S. The characteristic
cycle Z = Ch(!) is a semi-algebraic d-cycle in T* X, which is defined as follows.
The support supp(Z) =As. The smooth part A~ inherits an orientation from the
orientation of X , as in Section 2.1. Let now ~ E A~, and p = 7r(~) E X. The
multiplicity m€ Z of Z at ~ is given by:
m€Z = f(p) - x(H(0, f).
Exercises 2.5.l and 2.5.2 below may require some results of stratified Morse
theory, which we will state in the next lecture.