286 M. GRINBERG AND R. MACPHERSON, EULER CHARACTERISTICS(Recall that each standard pair (yt, Zt) must be transverse to S.) Then the push-
forward map H*(Yo, Zo; F)---> H*(Y 1 , Z 1 ; F) is an isomorphism.(b) The red and green walls of Y, or their intersections with any stratum
may touch, u~dergoing a Morse bifurcation, without a change in homology, as Ion~as the set Y is between the touching walls. More precisely, suppose S c X is a
stratum of codimension c < d, and U C X is a small open set with coordinates
(xi, ... ,xd): u ~]-1,l[d, such that snu = {x1 = ... =Xe = O}. Let
a(x1, ... , Xe) be a lmear form, such that d 0 a is generic in the sense of Section 2.3,
and v(xc+1, ... , xd-1) be a non-degenerate quadratic form. Assume (Y Z) is a
standard pair such that: 'y n U = { -E::::; Xd::::; a(Xi, ... , Xe)+ v(Xe+l>... , Xd-I) },
z n u = { y E Y I xd = -E},
where 0 < E « l. Fix a smooth cut-off function¢: [O, +oo[--+ [O, 1] satisfying:
(i) ¢(r) = 1, if r < 1/3;
(ii) ¢(r) = 0, if r > 2/3;
(iii) -10::::; ¢'(r)::::; 0, for all r.Fo~ x E U , let_ r(x) = Jxi + · · · + x~. Consider a new standard pair (Y', Z'),
which agrees with (Y, Z) outside of U, such that:
Y' nu= { -E + 2E. ¢(r(x))::::; Xd::::; a(x1, ... 'Xe)+ v(xe+l> ... 'Xd-1) },
Z' nu= { y E YI Xd = -E + 2E. ¢(r(x)) }.T~en the push-forward map H(Y, Z; F) ---> H(Y', Z'; F) is an isomorphism (see
Figure 2).
!
Green
Figure 2