We introduce several new families of relations in the mapping class groups of planar surfaces, each equating two products of right-handed Dehn twists. The interest of these relations lies in their geometric interpretation in terms of rational blowdowns of 4-manifolds, specifically via monodromy substitution in Lefschetz fibrations. The simplest example is the lantern relation, already shown by the first author and Gurtas ('Lantern relations and rational blowdowns ', Proc. Amer. Math. Soc. 138 (2010) 1131-1142 to correspond to rational blowdown along a −4 sphere; here we give relations that extend that result to realize the 'generalized' rational blowdowns of Fintushel and Stern ('Rational blowdowns of smooth 4-manifolds', J. Differential Geom. 46 (1997) 181-235) and Park ('Seiberg-Witten invariants of generalised rational blow-downs ', Bull. Austral. Math. Soc. 56 (1997) 363-384) by monodromy substitution, as well as several of the families of rational blowdowns discovered by Stipsicz, Szabó, and Wahl ('Rational blowdowns and smoothings of surface singularities',

We give formulae for the Ozsváth-Szabó invariants of 4-manifolds X obtained by fiber sum of two manifolds M 1 , M 2 along surfaces † 1 , † 2 having trivial normal bundle and genus g 1. The formulae follow from a general theorem on the Ozsváth-Szabó invariants of the result of gluing two 4-manifolds along a common boundary, which is phrased in terms of relative invariants of the pieces. These relative invariants take values in a version of Heegaard Floer homology with coefficients in modules over certain Novikov rings; the fiber sum formula follows from the theorem that this "perturbed" version of Heegaard Floer theory recovers the usual Ozsváth-Szabó invariants, when the 4-manifold in question has b 57R58; 57M99

We calculate the Heegaard Floer homologies$HF^+(M,s) for mapping tori M associated to certain surface diffeomorphisms, where s is any Spin^c structure on M whose first Chern class is non-torsion. Let gamma and delta be a pair of geometrically dual nonseparating curves on a genus g Riemann surface Sigma_g, and let sigma be a curve separating Sigma_g into components of genus 1 and g-1. Write t-gamma, t_delta, and t_sigma for the right-handed Dehn twists about each of these curves. The examples we consider are the mapping tori of the diffeomorphisms t_gamma^m circ t_delta^n for m,n in Z and that of t_sigma^{+-1}.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-31.abs.htm

Abstract. In this note we construct infinitely many distinct simply connected Stein fillings of a certain infinite family of contact 3-manifolds.

We make a detailed study of the Heegaard Floer homology of the product of a closed surface Σg of genus g with S 1 . We determine HF + (Σg × S 1 , s; C) completely in the case c 1 (s) = 0, which for g ≥ 3 was previously unknown. We show that in this case HF ∞ is closely related to the cohomology of the total space of a certain circle bundle over the Jacobian torus of Σg, and furthermore that HF + (Σg × S 1 , s; Z) contains nontrivial 2-torsion whenever g ≥ 3 and c 1 (s) = 0. This is the first example known to the authors of torsion in Z-coefficient Heegaard Floer homology. Our methods also give new information on the action of H 1 (Σg × S 1 ) on HF + (Σg × S 1 , s) when c 1 (s) is nonzero.