LECTURE 1. 2711.4. The Euler characteristic of a constructible function
Let X be a real algebraic manifold, and S be a stratification of X. We say that afunction f : X ___, Z is constructible with respect to S if it is constant on each stra-
tum of S, and that f is (simply) constructible if it is constructible with respect to
some stratification. We say that f is algebraically constructible if it is constructible
with respect to some semi-algebraic stratification.
The support supp(!) of a constructible function f is the closure of the set where
f is non-zero. We denote by :F x the set of all algebraically constructible functions
f : X ---+ Z with compact support. Note that :F x is an abelian group.
We now define a map x : :F x ---+ Z , called the Euler characteristic, or the integral
with respect to the Euler characteristic, as follows. Given a function f E :F x ,
subdivide the support supp(!) to a regular cell complex C, so that f is constant on
each cell. (Recall that a regular cell complex is a decomposition into finitely many
cells, such that each cell is homeomorphic to an open ball, and the closure of each
cell is a union of cells, homeomorphic to a closed ball.) Now set
awhere u runs over the cells of C.
This definition is convenient for simple computations, but there are several
problems with it. First, it uses the (difficult) result that any stratification of a
compact set can be subdivided to a regular cell complex. Second, we need to prove
that x(f) is independent of the particular subdivision chosen.
A more satisfactory definition, from the theoretical point of view, may be given
as follows. Given a subset A C X , write lA for the function which is equal to 1 on A
and to 0 on X\A. Then the Euler characteristic is the unique group homomorphism
x : :F x ___, Z, such that for any compact semi-algebraic subset A C X, we have:
x(lA) = x(A),where x(A) is the usual Euler characteristic of A.
Exercise 1.4.1. Show that the second definition gives a well defined functional x
(i.e., that such ax exists and is unique), and that it agrees with the first definition.
Exercise 1.4.2. Let f E :Fx, g E :Fy. Then the product f · g is in :FxxY· Show
that
xU · g) = x(f) · x(g).
Exercise 1.4.3. For a proper algebraic map m : X ---+ Y, define a push-forward
homomorphism m. : :Fx---+ :Fy, so that x(f) = x(m.f) for any f E :Fx.
Sometimes, we will need to consider the Euler characteristic of a general (notnecessarily algebraically) constructible function f : X ___, Z with compact support.
The definition of x(f) in this case is analogous. The only pitfall is that the set of
all such functions is not closed under addition.
We will be using the following shorthands. Whenever Z C Y C X is a pair of