Abstracts - Marco Golla

Abstracts

Line arrangements with odd multiplicities, arXiv:2403.17260.
We give restrictions on the weak combinatorics of line arrangements with singular points of odd multiplicity using topological arguments on locally-flat spheres in 4-manifolds. As a corollary, we show that there is no line arrangement comprising 13 lines and with only triple points.

(with M. Marengon) Splitting links by integer homology spheres, arXiv:2403.00064.
For every n at least 3 we construct 2-component links of in the (n+1)-sphere whose components are separated by a homology n-sphere but not by an n-sphere. We give a finer statement in dimension 4, where we find 2-component 2-links whose components are split by certain homology spheres but not by certain others.

(with L. F. Di Cerbo) The signature of geometrically decomposable aspherical 4-manifolds, arXiv:2306.15501.
We prove that for a geometrically decomposable (in the sense of Hillman) aspherical 4-manifold, the signature is bounded by the Euler characteristic (in fact, with a factor of 3). This proves a conjecture of Gromov for geometrically decomposable 4-manifolds. We also build some new examples of aspherical 4-manifolds with non-zero signature.

(with F. Ben Aribi, Sylvain Courte, and Delphine Moussard) Multisections of higher-dimensional manifolds, arXiv:2303.08779.
We study decompositions of higher-dimensional manifolds into 1-handlebodies, generalising work of Gay and Kirby in dimension 4, and of Rubinstein and Tillman in higher dimension. We focus on the case of quadrisections of 5-manifolds, for which we prove an existence result.

(with L. F. Di Cerbo) On the impossibility of complex-hyperbolic Einstein Dehn filling, arXiv:2111.12667.
Using the Hitchin–Thorpe inequality and an example of Hirzebruch, we show that the complex-hyperbolic Einstein Dehn filling compactification cannot be performed in dimension four.

(with L. Starkston) Rational cuspidal curves and symplectic fillings, arXiv:2111.09700.
We study symplectic fillings of rational cuspidal contact manifolds (introduced here) through Stein handlebodies and rational blow-downs. We give examples of such contact manifolds which are identifiable as links of normal surface singularities, other examples which admit no symplectic fillings, and further examples where the fillings can be fully classified.

(with K. Larson) 3-manifolds that bound no definite 4-manifold, arXiv:2012.12929.
We use correction terms in Heegaard Floer homology and a refinement of Elkie's theorem on short characteristic vectors in unimodular lattices to give a criterion for a 3-manifold not to bound any definite 4-manifold. We then produce a homology L(2,1), a connected sum of Seifert fibred spaces, satisfies the criterion.

(with F. Kütle) Symplectic isotopy of rational cuspidal sextics and septics, arXiv:2008.10923.
We study the symplectic isotopy problem of rational cuspidal curves of degree 6 and 7 in the complex projective plane. We prove that any such curve is symplectically equisingularly isotopic to a complex one.

(with P. Aceto, K. Larson, and A. Lecuona) Surgeries on torus knots, rational homology balls, and cabling, arXiv:2008.06760.
We classify all positive integral surgery on positive torus knots that bound rational homology balls. We also look at cabling slopes in relationship to surgery and to rational homology balls, Montesinos knots and links, and at trace embeddings in the complex projective plane. Proofs use Heegaard Floer homology, lattice embeddings, and a healthy does of Kirby calculus.

(with P. Feller) Non-orientable slice surfaces and inscribed rectangles, arXiv:2003.01590.
We prove that every locally 1-Lipshitz curve in the plane has an inscribed square and an inscribed rectangle of aspect ratio equal to the square root of 3. To prove it, we take inspiration from Hugelmeyer's ideas and we have fun playing around with locally flat surfaces in 4-manifolds.

(with J. Etnyre) Symplectic hats, arXiv:2001.08978.
A hat for a transverse knot in a symplectic cap of a contact 3-manifold is a symplectic surface in the cap whose boundary is the knot. We study existence, obstructions, and properties of hats, with an emphasis on transverse knots in the standard 3-sphere, and in caps that are obtained by blowing up the complement of a Darboux ball in the complex projective plane.

(with L. Starkston) The symplectic isotopy problem for rational cuspidal curves, arXiv:1907.06787.
We study symplectic rational cuspidal curves and their isotopies, focusing on curves in the complex projective plane. (Here a cusp is any irreducible complex curve singularity.) We give existence and uniqueness results for curves of degrees up to 7 and for curves with one cusp whose link is a torus knot.

(with C. Scaduto) On definite lattices bounded by integer surgeries along knots with slice genus at most 2, 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.
Erratum: in Table 1, the Brieskorn sphere –Σ(2,3,4) is associated to n = 3, while Σ(2,3,3) is not any manifold in the table.

(with D. Celoria and an appendix with A. Levine) Heegaard Floer homology and concordance bounds on the Thurston norm, 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, 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, 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) Surface singularities and planar contact structures, 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, arXiv:1610.10032.
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, arXiv:1607.08117.
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, arXiv:1509.08821.
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, arXiv:1505.07401.
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, arXiv:1503.05839.
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, arXiv:1607.06388.
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, arXiv:1509.07559.
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, arXiv:1309.3282.
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, arXiv:1412.0828.
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, arXiv:1201.5286.
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, arXiv:1505.07401.
We give a comparison between the EH contact invariant of Honda, Kazez and Matic and the LOSS^- invariant of Lisca, Ozsváth, Stipsicz and Szabó.