C".

2g their images on ~i). Lemma } shows that (we identify ~'l with From the construction of A' we know that A' ~ A ~ pQ, that is, ~I is diffeomorphic - to A ~ pQ ~ vP. ,2g. Clearly is diffeomorphic to A ~ pQ / ~P. But from Lemma 2 applied inductively we get that AII is an S2-bundle over S 2. ,2gc, (corresp. ~ , to (P ~ Q) ~ pQ / ~P. ,2gc, 2gc+V ) in A~II. Let ~ (corresp. ~ C, ~ = 2gc+l,... ,2gc+V. Because AII is simply connected and has an odd intersection form we have from results of Wall (see [8]) that ~ is diffeomorphic to ( P ~ Q) ~ pQ ~ v p ~ [2(gc-g)+9 ] (p ~ Q).

Z~(1)) 2 B{. - Since z~(1)(z~(1)) 2 = 0 [Recall that is a local equation of V O D f-l(o) has no multiple components f f2 = (z~C1))2_z~(1)(zi(1))2 is an invertible holomorphic B 1' in 28 function in some neighborhood of YI" zI = ~ zi(1) smaller]. z2 = ~ J z~ (I) • We take ! z 3 = z3(1) and choose Now l e t ~ = and we can identify B 1 fiB 1 with the function ~(Zl,Z2,Z3) E G3llZll < 2, lz21 < 2, Z = {(Zl,Z2,Z3) ~ ~3 z2-z3z 2 2I : i}, -- ~O where ~O,WI,~2 1 Iz31 < 7}, T = Z A ~. Identifying ~3 with ~2x~l we embed ~3 in ~p2x~I.