1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

284 M. GRINBERG AND R. MACPHERSON, EULER CHARACTERISTICS

Note that the pair (Y, Z) contains all the information necessary to recover the

stratification of Y, and the color scheme that comes with it.

We now sketch a proof of Theorem 1.1 in the case when Ji : X -+ Z is any

S-constructible function, and h = lY\z, for some standard pair (Y, Z) in X.

Let Y^0 = Y \{corner strata}. We can find a smooth function g: Y^0 -+ [O, 1],

such that g-^1 (0) ={the green walls}, g-^1 (1) ={the red walls}, and g restricted to

the open part of Y is stratified Morse with respect to S.

The proof of Theorem 1.1 for the functions Ji, h now comes from looking at

the Morse theory of g. This is in close analogy with the argument of Section 3.2.

Exercise. Furnish the missing details.

4.2. Proof of Theorem 1.1: The general case

We now turn to the general case of Theorem 1.1. We can assume without loss

of generality that the function h has compact support (we can always cut it off

outside of the support of h · h, which is compact). Let S2 be the set of all strata

of S2 contained in the support of h. For each stratum S E S2, we define a subset

S C X as follows.

Fix a Riemannian metric on X, and positive numbers 0 <Ed« Ed-1 « · · · «

Eo « 1 (d = dimX). For a point stratum S 0 = {p}, let So be the closed Ea-ball

around p.

Assume now S has been defined for any SE S2 with dimS < i (i = 1, ... ,d).

Let Xi-1 be the union of the s for all such s. Then, for an i-stratum Si E s2, we

let

Inductively, this defines S for every SE S2. Basically, we just thicken each stratum

up to a tubular neighborhood, in such a way that the strata of lower dimension

"get priority" over the higher ones.

Let ]2 : X -+ Z be the function which is equal to h ( S) on S, for every S E S2,

and to zero everywhere else.

Exercise 4.2.1. Show that replacing h with ] 2 will not change either side of the

equality in Theorem 1.1.

Given Exercise 4.2.1, and using linearity, we can reduce Theorem 1.1 to the

case where h = 15, for some SE S2. Let Y be the closure of S, and Z = Y \ S.

Exercise 4.2.2. Show that Y is a manifold with corners (of arbitrary codimension),

and that Z is a closed union of walls in Y.

The pair (Y, Z) isn't quite standard in the sense on Section 4.1. The last step
of the proof is to replace (Y, Z) by a standard pair (Y, Z), obtained by "smoothing
out" the corners of codimension more than 2. This places us in the situation of
Section 4.1, and the proof is complete.

Exercise 4.2.3. Furnish the missing details.