Abstracts - Marco Golla

Abstracts

(with C. Scaduto) On definite lattices bounded by integer surgeries along knots with slice genus at most 2
Available on the arXiv: arXiv:1806.11931.
We classify the positive definite intersection forms that arise from smooth 4-manifolds with torsion-free homology bounded by positive integer surgeries on the right-handed trefoil. A similar, slightly less complete classification is given for the (2,5)-torus knot, and analogous results are obtained for integer surgeries on knots of slice genus at most two. The proofs use input from Yang–Mills instanton gauge theory and Heegaard Floer correction terms.

(with D. Celoria) Heegaard Floer homology and concordance bounds on the Thurston norm (with an appendix with A. Levine)
Available on the arXiv: arXiv:1806.10562.
We give lower bounds on the Thurston norm on 2-component link complements; these bounds are invariant under strong concordance of the link. The main application is a concordance lower bound on the geometric winding number (or wrapping number) of a knot in S²×S¹. In the appendix, we discuss how similar techniques can be used to give lower bounds on the 0-shake-slice genus.

(with K. Larson) Linear independence in the rational homology cobordism group
Available on the arXiv: arXiv:1803.07931.
We study subgroups of the group of rational homology spheres modulo rational homology cobordism, using Heegaard Floer correction terms. In particular, we give sufficient conditions for a family of Z/2Z-homology spheres to generate a free subgroup.

(with A. Juhász) Functoriality of the EH class and the LOSS invariant under Lagrangian concordances
Available on the arXiv: arXiv:1801.03716.
We study the functoriality of the EH invariant of Honda–Kazez–Matić and of the LOSS invariant of Lisca–Ozsváth–Stipsicz–Szabó under Lagrangian cobordisms.

(with P. Ghiggini and O. Plamenevskaya) An obstruction to planarity of contact 3-manifolds
Available on the arXiv: arXiv:1708.04108.
We prove that if a contact 3-manifold admits an open book decomposition of genus 0, a certain intersection pattern cannot appear in the homology of any of its symplectic filling. We apply the obstruction to links of hypersurface singularities in complex dimension 3, Seifert manifolds, and integer homology balls. We also construct examples with arbitrary (finitely generated) fundamental groups.

(with P. Aceto and A. G. Lecuona) Handle decompositions of rational balls and Casson–Gordon invariants
Proceedings of the American Mathematical Society 146 (2018), no. 9, 4059–4072.
We give lower bounds on the minimal number of handles needed to construct rational homology balls with a given boundary. This in turn gives lower bounds on the number of bands needed to construct a ribbon disc for a ribbon knot in the 3-sphere.
(with M. Marengon) Correction terms and the non-orientable slice genus
Michigan Mathematical Journal 67 (2018), no. 1, 59–82.
We give lower bounds on the non-orientable slice genus (or 4-dimensional crosscap number) in terms of correction terms of surgeries. This is inspired by Batson's work. We compare our bound with the one given by Ozsváth–Stipsicz–Szabó, and define some new concordance invariants.
(with J. Bodnár and D. Celoria) A note on cobordisms of algebraic knots
Algebraic and Geometric Topology 17 (2017), no. 4, 2543–2564.
We study cobordisms of knots using correction terms in Heegaard Floer homology, along with properties of the concordance invariant nu+. We give emphasis on algebraic knots and, more generally, L-space knots, and algebraic cobordisms.
(with S. Behrens) Heegaard Floer correction terms, with a twist
Quantum Topology 9 (2018), no. 1, 1–37.
We use Heegaard Floer homology with twisted coefficients to define numerical invariants for arbitrary closed 3-manifolds equipped with a torsion spin^c structure, generalising the correction terms (or d-invariants) defined by Ozsváth and Szabó for integer homology 3-spheres and, more generally, for 3-manifolds with standard HF-infinity. Our twisted correction terms share many properties with their untwisted analogues. In particular, they provide restrictions on the topology of 4-manifolds bounding a given 3-manifold.
(with B. Martelli) Pair of pants decompositions of 4-manifolds
Algebraic and Geometric Topology 17 (2017), no. 3, 1407–1444.
We study pair of pants decompositions of 4-manifolds, generalising a construction due to Mikhalkin. We construct some nontrivial examples, and in particular we prove that any finitely presented group is the fundamental group of a closed 4-manifold admitting a pants decomposition.
(with P. Aceto and K. Larson) Embedding 3-manifolds in spin 4-manifolds
Journal of Topology 10 (2017), no. 2, 301–323.
We give obstructions and constructions of embeddings of rational homology balls into spin 4-manifolds, focusing on connected sums of S²×S². Our favourite examples are surgeries along knots in the 3-sphere, lens spaces, and Seifert fibred spaces. We mostly give obstructions using the Rokhlin invariant and the 10/8 Theorem.
(with P. Aceto) Dehn surgeries and rational homology balls
Algebraic and Geometric Topology 17 (2017), no. 1, 487–527
We study which Dehn surgeries along knots bound rational homology balls, giving restrictions on the possible surgery slopes and constructions for some families of surgeries along torus knots. The main tools are Heegaard Floer correction terms, Donaldson's theorem, and manipulations of plumbing diagrams.
(with J. Bodnár and D. Celoria) Cuspidal curves and Heegaard Floer homology
Proceedings of the London Mathematical Society 112 (2016), no. 3, 512–548
We give bounds on the gap functions of the singularities of a cuspidal plane curve of arbitrary genus. We apply these inequalities to unicuspidal curves whose singularity has one Puiseux pair: we prove some finiteness results, we construct infinite family of examples, and in some cases we give an almost complete classification.
(with P. Lisca) On Stein fillings of contact torus bundles
Bulletin of the London Mathematical Society 48 (2016), no. 1, 19–37
We construct tight contact structures on some torus bundles over the circle and we study their Stein fillings, up to diffeomorphism. We classify these fillings, showing that uniqueness holds in some cases; we also provide examples where uniqueness does not hold.
Ozsváth–Szabó invariants of contact surgeries
Geometry and Topology 19, no. 1, (2015) 171–235
We give the computation of the Ozsváth–Szabó contact invariant for positive contact surgeries in the 3-sphere in terms of the classical invariants of the Legendrian knot, and tau and nu (or tau and epsilon) of the underlying topological type.
Comparing invariants of Legendrian knots
Quantum Topology 6 (2015), no. 3, 365–402
We give a comparison between the EH contact invariant of Honda, Kazez and Matic and the LOSS^- invariant of Lisca, Ozsváth, Stipsicz and Szabó.