278 M. GRINBERG AND R. MACPHERSON, EULER CHARACTERISTICSExercise 2.5.1. Show that the multiplicity m~Z is equal to x(f'), where f' is the
function equal to f on N n B E n g-^1 ] - oo, 6[ and to zero elsewhere (N, BE, g, and
b are as in Section 2 .4).
Exercise 2.5.2. Show that Ch(!) is , indeed, a cycle, i.e., that it has no boundary
of dimension d - 1.
Exercise 2.5.3. Show that Ch(!) only depends on the function f, and not on the
stratification S.
Exercise 2.5.4. Show that f r--+ Ch(!) gives a bijection (in fact, a group homo-
morphism) between the set of all S-constructible functions f : X ---> Z, and the set
of all d-cycles in T * X supported on As.
Exercise 2.5.5. Show that f r--+ Ch(!) gives a bijection between :F x and the set
of all d-cycles Z in T X , such that supp(Z) C T X is a Lagrangian subvariety,
invariant under the scalar action of JR+. (By a Lagrangian subvariety we mean the
closure of a semi-algebraic Lagrangian submanifold.)
2.6. Signs
The statement of Theorem 1.1 is now a lmost complete. It remains to make two
comments about signs. First, in computing the intersection number, we orient T * X
by the orientation of the base, followed by the orientation of the fiber. (This may
differ from the symplectic orientation.)
Second, the antipodal map -1 : T X ---> T X induces a map on d-cycles in the
obvious way. E.g., it takes the zero section into itself, and each cotangent fiber into
( -1 )d times itself.
With this, the statement of Theorem 1.1 is complete. We'll take up the proof
in the next le cture.