Corks, involutions, and Heegaard Floer homology- Abhishek Mallick (Michigan State)
Abstract: We introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution of a 3-manifold over any homology ball that it may bound (rather than a particular contractible manifold). We utilize the formalism of local equivalence coming from involutive Heegaard Floer homology. As an application, we define a modification of the integer homology cobordism group which takes into account involutions acting on homology spheres, and prove that this group admits an infinite rank subgroup generated by corks. Using our invariants, we also establish several new families of (strong) corks (This is joint work with Irving Dai and Matthew Hedden). This talk will be complementary to the talk given by Irving Dai (in the "Floer theory and low-dimensional topology" topic group).
A relative invariant of smooth 4-manifolds- Hyunki Min (Georgia Tech)
Abstract: We define a polynomial invariant of a smooth and compact 4-manifold with connected boundary by modifying an invariant of closed 4-manifolds from Heegaard Floer homology. Using this invariant, we show that there exist infinitely many exotic fillings of 3-manifolds with non-vanishing contact invariant. This is a joint work with John Etnyre and Anubhav Mukherjee.
Standardizing Some Low Genus Trisections- Jesse Moeller (University of Nebraska-Lincoln)
Abstract: Trisections are a novel way to study smooth 4-manifold topology reminiscent of Heegaard splittings of a 3-manifolds; a surface, together with families of embedded curves, determines a 4-manifold up to diffeomorphism. Given a specific genus, we would like to know which 4-manifolds have trisection diagrams inhabiting a surface with this genus. For genus one and two, this is known. In this talk, we will introduce a family of seemingly complicated genus three trisection diagrams and demonstrate that they are, in fact, connected sums of well understood diagrams.
Symplectic 4-Manifolds on the Noether Line and between the Noether and Half Noether Lines - Sümeyra Sakalli (MPIM Bonn)
Abstract: It is known that all minimal complex surfaces of general type have exactly one (Seiberg-Witten) basic class, up to sign. Thus, it is natural to ask if one can construct smooth 4-manifolds with one basic class. First, Fintushel and Stern built simply connected, spin, smooth, nonsymplectic 4-manifolds with one basic class. Next, Fintushel, Park and Stern constructed simply connected, noncomplex, symplectic 4-manifolds with one basic class. Later Akhmedov constructed infinitely many simply connected, nonsymplectic and pairwise nondiffeomorphic 4-manifolds with nontrivial Seiberg-Witten invariants. Park and Yun also gave a construction of simply connected, nonspin, smooth, nonsymplectic 4-manifolds with one basic class. All these manifolds were obtained via knot surgeries, blow-ups and rational blow-downs. In this talk, we will first review some main concepts and recent techniques in symplectic 4-manifolds theory. Then we will construct minimal, simply connected and symplectic 4-manifolds on the Noether line and between the Noether and half Noether lines by the so-called star surgeries, and by using complex singularities. We will show that our manifolds have exotic smooth structures and each of them has one basic class. We will also present a completely geometric way of constructing certain configurations of Kodaira’s singularities in the rational elliptic surfaces, without using any monodromy arguments.
Ribbon homology cobordisms- Mike Wong (Louisiana State University)
Abstract: A cobordism between 3-manifolds is ribbon if it has no 3-handles. Such cobordisms arise naturally from several different topological and geometric contexts. In this talk, we describe a few obstructions to their existence, from Thurston geometries, character varieties, and instanton and Heegaard Floer homologies, and some applications. This is joint work with Aliakbar Daemi, Tye Lidman, and Shea Vela-Vick.