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Orlik and P. Wagreich, ‘Isolated singularities of algebraic surfaces with C∗ action’, Ann. of Math. 93 (1971) 205–228. 20. P. Orlik and P. Wagreich, ‘Algebraic surfaces with k ∗ –action’, Acta Math. 138 (1977) 43–81. ´th and Z. Szabo ´ , ‘Absolutely graded Floer homologies and intersections forms for four-manifolds 21. P. Ozsva with boundary’, Adv. Math. 173 (2003) 179–261. 22. H. Park, J. Park and D. 0273. 23. J. Park, ‘Seiberg–Witten invariants of generalized rational blow-downs’, Bull. Austral.

This gives the desired boundaries for all graphs of type M. 4 (Graphs in the families W and N ). A similar construction, starting with C the union of four lines L1 , L2 , L3 , L4 in general position in CP2 , yields rational homology ball smoothings for the triply infinite families of type W and N . For the (3,3,3) type W, blow up six times, giving the following. e4 e1 ◦ • L3 e2 • • +1 • • L2 L1 L4 • ◦ e5 • e3 ◦ e6 Blowing up p additional times between L4 and e6 and successor (−1)-curves, q times between L3 and e5 , and r times between L2 and e4 , and then removing the three (−1)-curves, gives a Γ which is easily checked to be dual to Γp,q ,r .

E4 , G1 , . . , Gr , H1 , . . , Hq }). 516 ´ I. STIPSICZ, ZOLT AN ´ SZAB O ´ AND JONATHAN WAHL ANDR AS Figure 26. The embedding of the graph Λp,q ,r for p, r 1 and q 0; here C = E1 − H1 , D1 = Hq +1 − G1 − E3 , D2 = Gr − F1 − E2 , P = E2 − E1 − E4 − Fp−1 − . . − F1 , Q = E4 − E1 − E3 − Hq +1 − . . − H1 , R = E3 − E1 − E2 − Gr − . . − G1 . Figure 27. The embedding of the graph Λ0,q ,r for r 1 and q 0; C = E1 − H1 , D = Hq +1 − E3 − G1 , Q = E3 − E1 − E4 − E2 − Hq +1 − . . − H1 , R = E4 − E1 − E3 − E2 − Gr −1 − .

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