276 M. GRINBERG AND R. MACPHERSON, EULER CHARACTERISTICS(a) For any point a E A, and a sequence {bi } (i = 1, 2, ... ) of points in B,assume that bi ---+ a, and there is a limit L = lim Tb; B C TalRn. Then we have:
L :J TaA.(b) For any point a EA and sequences {ai EA} and {bi EB}, assume that:
ai ---+ a and bi ---+ a. Write bi ai for the secant line through bi and ai. Think of bi aias a subspace of Tb, ]Rn. Assume that there are limits L = lim Tb, B C TalRn, and
K = limbi ai C TalRn. Then we have: L :J K.
Exercise 2.2.1. Show that Whitney conditions are invariant under diffeomor-
phisms on ]Rn, so they are well defined when A and B are submanifolds of any
smooth manifold X.Exercise 2.2.2. Show that condition (b) implies condition (a).Exercise 2.2.3. Return to the Whitney cusp example of Exercise 1.2, but now
considered over the complex numbers. I.e., setZ = {( x, y, z ) E C^3 I y^2 = x^2 (z^2 - x)}.
Consider the decomposition of Z into two manifolds: Z = A U B, where A is the
z-axis, and B = Z \A. Show that Whitney condition (a) holds for the pair (A, B),
but that condition (b) does not. (Our reason for passing to the complexes is just
to make B connected.)One may ask why we need to talk about condition (a) when it is implied by
condition (b). One reason is historic. Whitney stated condition (a) first, then
discovered the Whitney cusp, and realized that condition (a) alone was not enough
to ensure local topological triviality along the strata. Another reason is that it is
often convenient to verify (a) first, then to verify (b) in the presence of (a), which is
simpler. Yet another reason is that condition (a) has the following neat microlocal
formulation.Exercise 2.2.4. Let X be a smooth manifold, and S = {Si } be a decomposition
of X into submanifolds, satisfying conditions (1) and (2) of Section 1.2. Then
Whitney condition (a) holds for each pair (Si,Sj) with Si C Sj, if and only if As
is closed in T* X.
Exercise 2.2.5. Can you find a microlocal formulation of condition (b)? (There
is a formulation in terms of data on T* X, but it uses the canonical 1-form, so it is
not in purely symplectic language.)
2.3. Generic covectors
In the next two sections we will talk about some of the main notions of stratified
Morse theory. Here we discuss generic conormal vectors to a stratification.
Let X be a smooth manifold, S a stratification of X, S a stratum of S , and
As C T* X the conormal bundle to S. A covector ~ E As is called generic with
respect to S if it does not annihilate any limit of tangent spaces to any stratum
other than S.