On the Donaldson polynomials of elliptic surfaces 633

This implies the first part of the statement. Next, choose аеЯгСМг; Z) with а.афО. Then

7c ( ^2 ) ( a^ " ) =^ ( a . ar = iAo(iV2)-WM2)(a^"),

while

) 'c + l ( ^'2 ) ( s^ / ^a^ " ) =^тJ ; j ( a . a ) " = ^A4(JV,)(s^Я) ..Ad(M,)(a^").

The conclusion is clear. П

Remark . An argument similar to that given in the proof of 2.1 shows that ^2{N2){sJ) generates HF,{dN2).

Proposition 2.2. Let X be a smooth simply connected 4-manifold with Ьг (^) odd, b2{X)^3. Suppose there is an orientation-preserving embedding N2<=^X, and denote by s and f the images inside ЯзСХ; Z) of the classes of a section and a fiber. Then, if d denotes the degree ofydX), for all c^O and any collection ai,. . ., (Xd-4eH2{X; Ж) of classes orthogonal to s andf

Proof The proof is straightforward. Let d be the degree ofydX). Recall [FM] that, for all ft ^0,

y , ( X ) ( sV^ai , . . .,ad_4)=T324h Vc+л(^)(5^/^al,. . .,аа-4,4,. • .,4).

where X = X# ftCIP^ and ei,. . ., e^ are the Poincaré duals of the exceptional classes. Since the connected sums may be taken inside X\N2, if h is large enough, so that с + ft is in the stable range, we have

7e ( X ) ( s^^ai , . . .,ad-4) while

x ( ai , . . .,ad-4,4,. • -,4)-

The conclusion follows immediately from Lemma 2.L П

Corollary 2.3. Let X be a smooth simply connected 4-manifold with b2{X)^3 odd, and let qx be its intersection form. Suppose there is кеН2(Х; Ж) such that the Donaldson invariants ofX are polynomials in к and qx- Moreover, suppose there is an orientation-preserving embedding N2^X with the image of ЯзСЛГа; Z) orthogonal to k. Then, for all c^l and /^0,

a'i { X ) =^ar4X )