1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

280 M. GRINBERG AND R. MACPHERSON, EULER CHARACTERISTICS

Assuming now that g is Morse, we find that

Ax nAx =Ax n {dg} = l_)-1)>-(P),

p

where p runs over the critical points of g.

Now, it is a standard fact of classical Morse theory that I:P(-1)>.(p) = x(X).

The proof is based on the following two results.

Theorem 3.1.4. Let g : X ---t JR be a proper Morse function. Assume g has

no critical values in the interval [a, b] E JR. Then g-^1 [a, b] is diffeomorphic to

g -^1 (a) x [a, b]. It follows that X::;a ~ X::;b, where X::; = g-^1 ] - oo, ].

Theorem 3.1.5. Let g : X ---t JR be a proper Morse function, a E JR a critical

value of g, and p E X the unique critical point with g(p) =a. Let d = dimX and

>.. = A.(p). Fix a small number E > 0. Then the set X::;a+< is obtained from the set

X<a-< by gluing on the product D>. x Dd->. along the subset an>-x Dd->. (here D n

de-:,wtes the standard n-ball). It follows that x(X::;a+<) = x(X::;a-<) + (-1)>..

We will say that the pair (D>. x Dd->., an>-x Dd->.) is the Morse data of g at

p. The proof of Theorem 1.1 in the case Ji = h = 1 is now complete.

3.2. Stratified Morse theory

Our next step is to prove Theorem 1.1 when h = 1, and Ji : X ---t Z is any alge-

braically constructible function with compact support. The proof will be parallel

to the argument in Section 3.1, with stratified Morse theory replacing the classical

one.

We fix a semi-algebraic stratification S of X, so that Ji is S-constructible. Let

g : X ---t JR be a proper smooth function. A point p E X is called stratified critical

for g if it is critical for the restriction gls, where Sis the stratum of S containing

p. In other words, p is critical if and only if dpg E As.

We say that p is a stratified Morse critical point for g if it is a Morse critical

point for gls, and the covector dpg is generic. Also, we say that g is (stratified)

Morse if all of its critical points are.

Exercise 3.2.1. Show that g is a stratified Morse function if and only if the

intersection of {dg} and As is transverse (i.e., {dg} meets As only along A~, and

all the intersection points are transverse).

We can now state the analog of Theorem 3.1.4.

Theorem 3.2.2. Let (X, S) be as above, and let g : X ---t JR be a proper stratified

Morse function. Assume g has no (stratified) critical values in the interval [a, b] E

JR. Then g-^1 [a,b] is homeomorphic to g-^1 (a) x [a,b] in a stratum-preserving way

(both spaces inherit their stratifications from X).

On the level of Euler characteristics this implies that

x(X::;a; Ji)= x(X::;b; Ji).

To state the analog of Theorem 3.1.5, we need to define the normal and tan-
gential Morse data at a stratified Morse critical point p of g. The tangential Morse

data of g at pis just the Morse data for gls at p. The normal Morse data is defined