272 M. GRINBERG AND R. MACPHERSON, EULER CHARACTERISTICS1.5. Homology n-product of cycles
The next ingredient of Theorem 1.1 we need to explain is the n-product of semi-
algebraic cycles.
Let M be an oriented real algebraic manifold (in the situation of Theorem 1.1
M = T* X). A semi-algebraic i-cycle Z in M is a stratified, closed semi-algebraicsubset supp(Z) c M of dimension i, plus an orientation and a multiplicity on each
i-stratum, such that the cycle relation holds along each ( i - 1 )-stratum. We assume
that the stratification of supp(Z) is semi-algebraic, and that supp(Z) is of pure
dimension i, i.e., that it is equal to the closure of its i-strata. The multiplicity of
an i-stratum is an integer. As usual, the pair (orientation, multiplicity) is defined
up to a simultaneous sign change. The cycle relations mean that Z has no (i - 1)-
dimensional boundary (a precise definition can be given by triangulating supp( Z)).
They imply that Z represents a class in H i ( M, {the part near infinity}). We con-
sider two i-cycles equal if they differ by restratifying the support, or by adding
some i-strata with multiplicity zero.
Assume we have an i 1 -cycle Z 1 and an i2-cycle Z2, with i 1 + i 2 = dim M.
Assume, in addition, that the intersection supp(Z 1 ) n supp(Z2) is compact. Thenwe may define an integer Z 1 nz 2 , called the homology n-product, or the intersection
product of Z 1 and Z2. To do this, we move Z 1 into general position with Z 2 by an
isotopy 'ljJ : [O, 1 J x M ---> M which is trivial outside of some compact set. The general
position requirement is that 'ljJ 1 (S 1 ) must be transverse to S2, for any stratum S 1
of supp(Z 1 ) and any stratum S2 of supp(Z2). After this, we setZ1 n Z2 = L m ,μ!'(p)Z1 · mpZ2 ·sign.
p
Here p runs over the (necessarily finite) intersection 'ljJ 1 (supp(Z 1 )) n supp(Z 2 ), the
numbers m,μ 1 1(p)Zi and mpZ2 are the multiplicities of Z 1 and Z 2 at '!/J1^1 (p) and p ,
respectively, and the sign is computed in the usual way.Exercise 1.5.1. Show that Z 1 n Z2 is independent of the choice of the isotopy 'l/J.Exercise 1.5.2. Prove Theorem 1.1 in the case when Xis compact and Ji = f2 =
1. (The characteristic cycle of the identity function is the zero section TxX c
T*X.)
1.6. The characteristic cycle in dimension one
The last unexplained ingredient of Theorem 1.1 is the characteristic cycle of a
constructible function. We will give a general definition in the next lecture. Here,we only discuss the case X = ~^1. This is the largest example where dim T * X :::; 3,
and one can draw a realistic picture of the cotangent bundle.Let S be a stratification of X = ~^1 , and f : X ---> Z be a function constructible
with respect to S. Let k be the number of point strata of S , so that the totalnumber of strata is 2k + 1. The characteristic cycle Z = Ch(!) is defined as
follows. The support supp(Z) is the union of the zero section TxX c T*'X and the
fibers r;x c T* X, for all the point strata p of S. Note that supp(Z) is naturally
stratified with 4k + 1 strata. Both the zero section and each of the cotangent fibers
is canonically identified with ~, and we use the positive direction of the real line to
orient the 1-strata of supp(Z).