Thomas E. Mark

Product formulae for Ozsváth-Szabó 4-manifold invariants

We give formulae for the Ozsvath‐Szabo invariants of 4‐manifolds X obtained by fiber sum of two manifolds M1 , M2 along surfaces U1 , U2 having trivial normal bundle and genus g 1. The formulae follow from a general theorem on the Ozsvath‐ Szabo 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 Ozsvath‐Szabo invariants, when the 4‐manifold in question has b C 2. The construction allows an extension of the definition of Ozsvath‐Szabo invariants to 4‐manifolds having b C D 1 depending on certain choices, in close analogy with Seiberg‐Witten theory. The product formulae lead quickly to calculations of the Ozsvath‐Szabo invariants of various 4‐manifolds; in all cases the results are in accord with the conjectured equivalence between Ozsvath‐Szabo and Seiberg‐Witten invariants.

Heegaard Floer homology of certain mapping tori

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 γ and δ be a pair of geometrically dual nonseparating curves on a genus g Riemann surfaceg, and let σ be a curve separatingg into components of genus 1 and g −1. Write t, t�, and tfor the right-handed Dehn twists about each of these curves. The examples we consider are the mapping tori of the diffeomorphisms t m ◦ t n for m, n ∈ Z and that of t ±1 � . AMS Classification 57R58; 53D40

Knotted surfaces in 4-manifolds

Abstract. Fintushel and Stern have proved that if is a symplectic surface in a symplectic 4-manifold such that has simply-connected complement and nonnegative self-intersection, then there are infinitely many topologically equivalent but smoothly distinct embedded surfaces homologous to . Here we extend this result to include symplectic surfaces whose self-intersection is bounded below by , where g is the genus of . We make use of tools from Heegaard Floer theory, and include several results that may be of independent interest. Specifically we give an analogue for Ozsváth–Szabó invariants of the Fintushel–Stern knot surgery formula for Seiberg–Witten invariants, both for closed 4-manifolds and manifolds with boundary. This is based on a formula for the Ozsváth–Szabó invariants of the result of a logarithmic transformation, analogous to one obtained by Morgan–Mrowka–Szabó for Seiberg–Witten invariants, and the results on Ozsváth–Szabó invariants of fiber sums due to the author and Jabuka. In addition, we give a calculation of the twisted Heegaard Floer homology of circle bundles of “large” degree over Riemann surfaces.