1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 5. 289

5.3. Euler characteristics

Let F be a Fary functor on (X, S). Given a standard pair (Y, Z), let hi(Y, Z; F)

be the rank of the abelian group H i(Y, Z; F). A constructible function f = xF :

X ----+ Z, called the Euler characteristic of F, is defined by:

Exercise 5.3.1. Let (Y, Z) be a standard pair in X.. Show that

Exercise 5.3.2. Show that every function constructible with respect to S. is equal

to xF, for some F.

We now have a diagram:

{Fary functors}

l x 1

{Constructible functions}

Ch

-----+ {Lagrangian cycles}

The objects in the left column live on X, and the objects in the right column live
on T * X. The map Ch is an isomorphism (see Exercises 2.5.4, 2.5.5). An important
question is now the following: can we put something in place of the question mark,
so there would exist a natural isomorphism Ch, and the diagram would commute?
In other words, can we think of Fary functors as microlocal objects?

If we replace Fary functors by perverse sheaves (see Section 5.6 below), then

the answer is "yes." In an upcoming paper by S. Gel'fand , R. MacPherson, and K.

Vilonen [GMV], a microlocal version of perverse sheaves is constructed, ' which can

take the place of the question mark.

5.4. Poincare-Verdier duality

If we work with coefficients in Q, instead of Z , then the category of Fary functors

has a remarkable ( contravariant) involution F r---> F*, called the Poincare-Verdier

duality. It is defined by:

Hi(Y,Z;F*) = Hd-i(Y,Z;F)*,

where d = dimX, Z = 8Y \ Z (the red part of Y), and the upper *denotes the

vector space dual.

Exercise 5.4.1. Complete the definition of F * by specifying the structure maps.
Show that Verdier duality is an anti-equivalence on the category of Fary functors,
and that the square of it is the identity.

Exercise 5.4.2. Via the diagram of Section 5.3, Verdier duality induces maps on
both constructible functions and Lagrangian cycles. Compute these maps.