LECTURE 3. 281as the pair(N n B, n g-^1 [-8, +oo[N n B, n 9 -^1 (-8)),
where the notation is as in Section 2.4. Note that the set N n B, n g-^1 (-8) is just
the half-link of -dpg· (This is why we have the antipodal map in the statement of
Theorem 1.1.) The main result of stratified Morse theory is now the following.Theorem 3.2.3. Let g : X---+ JR be a proper (stratified) Morse function. Assumea E JR is a critical value of g, and p EX is the unique critical point with g(p) =a.
Write (YN, ZN) and (Yr, Zr) for the normal and tangential Morse data of g at p.
Then the product(Y, Z) = (YN, ZN) x (Yr, Zr)= (YN x Yr, YN x Zr U ZN x Yr)
gives Morse data for g at p. In other words, the set X::;a+' may be obtained, up to
a stratum-preserving homeomorphism, by gluing Y to X::;a-E along Z.We are being a bit imprecise here about what we mean by a stratum-preserving
homeomorphism. The set Y has a natural decomposition according to the strata
of S near p, and it is this decomposition that will be respected by the gluing
in Theorem 3.2.3. (One would have to refine this decomposition if one needs a
stratification of Y.)
On the level of Euler characteristics, Theorem 3.2.3 implies thatx(X::;a+E> X::;a-E; Ji)= x(YN, ZN; Ji). x(Yr, Zr; 1).
Note that the first term of the product in the right-hand side is the multiplicity ofCh(f 1 ) at -dpg (cf. Exercise 2.5.1), and that the second term is just ±1.
The proof of Theorem 1.1 in the case h = 1 is now just a matter of putting
things together. In order to compute the intersection (-l)*Ch(f 1 ) n Ch(h) =
( -1) * C h(f 1) n Ax, we may replace the zero section Ax by the graph of dg for
a suitable Morse function g : X ..:..., R It is enough to assume that g is proper,non-negative, and that g-^1 [0, 1] contains the support of fi. Then both sides of the
equation.xU1 x h) = (-l)*Ch(f1) n Ch(h)
become sums over the critical points p of g , with g(p) :::; 1, of products
{multiplicity}· {sign}.We have already seen that the multiplicities are the same for both sides. We leave
it to the reader to verify that so are the signs.
3.3. Comments
You may wonder about the level of difficulty of Theorems 3.2.2 and 3.2.3. They cer-
tainly sound very natural. However, they are quite difficult to prove. The problem
is that stratification theory is an odd mixture of C^00 and c^0 objects. All points on
the same stratum have neighborhoods that look the same topologically, but noth-
ing like this is true in any smooth sense; so it is not at all obvious that Whitney
conditions assure enough uniformity along the strata to make Theorems 3.2.2 and
3.2.3 work. Theorem 3.2.2 is due to Thom and Mather ([Th], [Mal], [Ma2]), and
Theorem 3.2.3 is due to Goresky and MacPherson [GM]. The introduction to the
book [GM] has a nice set of figures illustrating Theorem 3.2.3.