1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 1

1.1. The Centerpiece Theorem

The main result of these lectures is the following.

Theorem. Let X be an oriented real algebraic manifold, and Si, S 2 two semi-

algebraic Whitney stratifications of X. Assume that each stratum of S i is transverse

to each stratum of S2. Let Ji, h be two functions X ---+ Z , such that Ji is constant

on each stratum of S i (i = 1, 2). Assume that the product Ji · h : X ---+ Z has

compact support. Then the intersection product (-l)*Ch(fi) nCh(h) E Z is well

defined, and we have:

x(fi · h) = (-l)*Ch(fi) n Ch(h).

Here x(fi · h) is the Euler characteristic of the function Ji · h, the symbol Ch(fi)

denotes the Lagrangian characteristic cycle of f i in T * X, and -l : T* X ---+ T* X

is the antipodal map.

The theorem in this form is a mild extension of a result of Brylinski-Dubson-

Kashiwara [BDK]. A detailed treatment of this and related results, as well as a note

on the history of the subject, can be found in [KS, Chapter 9]. In t he rest of this

lecture we will be introducing the different ingredients of this theorem.

1.2. Stratifications

Let X be a smooth manifold, Z C X a closed subset. A (Whitney) stratification S
of Z is a decomposition

Z= LJ S

SES
of Z into a disjoint union of finitely many subsets, called the strata of S , such that:
(1) .each stratum is a connected, locally closed smooth submanifold of X;
(2) the closure of each stratum is a union of strata;

(3) whenever Si, Sj are two strata with Si c Sj, Whitney conditions (a) and

(b) hold for the pair (Si , Sj).

We will state the Whitney condit ions in the next lecture. For now, we just say

that these are certain regularity conditions on the behavior of the larger stratum

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