arXiv subjects

Paolo Aceto

Publications and source records attributed to Paolo Aceto.

At least 19 recordsLinked to original sources

arXiv2026

Accessing Chern Numbers From Bloch Eigenstates Singularities

We propose two methods to extract the Chern numbers of the Bloch bands of a two dimensional insulator, and apply them to a photonic lattice experiment. These two methods require the knowledge of the complex-valued components of the Bloch eigenvectors, or their ratios. Unlike other tomography methods relying on the approximate reconstruction of the Berry curvature and its integration over the Brillouin zone, our methods boil down to the observation of phase vorticities in the eigensates structure. Those measurements, robust to experimental noise, yield exactly quantized values. One of the two methods exploits the non-normalization of the eigenmodes, making it particularly relevant for classical wave systems.

cond-mat.mes-hall↗

arXiv2026

Topological line arrangements with high multiplicities

We investigate constraints on the existence of topological and smooth realisations of combinatorial line arrangements and $(n_k)$-configurations in the complex projective plane. By replacing complex lines with locally-flatly or smoothly embedded 2-spheres, we explore the extent to which classical geometric results, such as Hirzebruch's inequality, persist in the topological or smooth category. We introduce two classes of special line arrangements that we call odd and even. We provide constraints for any smoothly realised, non-trivial, odd arrangement via Furuta's 10/8-Theorem. By looking at branched double covers and using the G-signature theorem, we study topologically realised, non-trivial, even arrangement. Finally, we establish a new lower bound for $(n_k)$-configurations, showing that for any topologically realised configuration we have $n \geq k^2-5$, which implies the non-existence of topological realisations for finite projective planes.

math.GT↗

arXiv2026

Smooth Realizations of Line Configurations

We study the problem of realizing line configurations as collections of 2-spheres smoothly embedded in the complex projective plane. Building upon prior work by Ruberman and Starkston on topological realizations, we establish a stronger obstruction in the smooth category. Our proof relies on lattice-theoretic arguments based on Donaldson's diagonalization theorem.

math.GT↗

arXiv2024

A survey on embeddings of 3-manifolds in definite 4-manifolds

This article presents a survey on the topic of embedding 3-manifolds in definite 4-manifolds, emphasizing the latest progress in the field. We will focus on the significant role played by Donaldson's diagonalization theorem and the combinatorics of integral lattices in understanding these embeddings. Additionally, the article introduces a new result concerning the embedding of amphichiral lens spaces in negative-definite manifolds.

math.GT↗

arXiv2023

Slice obstructions from genus bounds in definite 4-manifolds

We discuss an obstruction to a knot being smoothly slice that comes from minimum-genus bounds on smoothly embedded surfaces in definite 4-manifolds. As an example, we provide an alternate proof of the fact that the (2,1)-cable of the figure eight knot is not smoothly slice, as shown by Dai--Kang--Mallick--Park--Stoffregen in 2022. The main technical input of our argument consists of gauge-theoretic obstructions to smooth small-genus surfaces representing certain homology classes in $\mathbb{CP}^2\#\mathbb{CP}^2$ proved by Bryan in the 1990s.

math.GT↗

arXiv2022

Handle decomposition complexity and representation spaces

We prove that there are homology three-spheres that bound definite four-manifolds, but any such bounding four-manifold must be built out of many handles. The argument uses the homology cobordism invariant $\Gamma$ from instanton Floer homology.

math.GT↗

arXiv2022

Definite fillings of lens spaces

This paper considers the problem of determining the smallest (as measured by the second Betti number) smooth negative-definite filling of a lens space. The main result is to classify those lens spaces for which the associated negative-definite canonical plumbing is minimal. The classification takes the form of a list of 10 "forbidden" subgraphs that cannot appear in the plumbing graph if the corresponding plumbed 4-manifold is minimal. We also show that whenever the plumbing is minimal any other negative-definite filling for the given lens space has the same intersection form up to addition of diagonal summands. Consequences regarding smooth embeddings of lens spaces in 4-manifolds are also discussed.

math.GT↗

arXiv2020

Non-simply connected symplectic fillings of lens spaces

We prove results exploring the relationship between the fundamental group and the second Betti number of minimal symplectic fillings of lens spaces. These results unify and generalize several disparate facts appearing in the literature. The Fibonacci numbers make a cameo appearance.

math.GT↗

arXiv2020

Surgeries on torus knots, rational balls, and cabling

We classify which positive integral surgeries on positive torus knots bound rational homology balls. Additionally, for a given knot K we consider which cables K(p,q) admit integral surgeries that bound rational homology balls. For such cables, let S(K) be the set of corresponding rational numbers q/p. We show that S(K) is bounded for each K. Moreover, if n-surgery on K bounds a rational homology ball then n is an accumulation point for S(K).

math.GT↗

arXiv2020

Isotopy and equivalence of knots in 3-manifolds

We show that in a prime, closed, oriented 3-manifold M, equivalent knots are isotopic if and only if the orientation preserving mapping class group is trivial. In the case of irreducible, closed, oriented $3$-manifolds we show the more general fact that every orientation preserving homeomorphism which preserves free homotopy classes of loops is isotopic to the identity. In the case of $S^1\times S^2$, we give infinitely many examples of knots whose isotopy classes are changed by the Gluck twist.

math.GT↗

arXiv2020

Branched covers bounding rational homology balls

Prime power fold cyclic branched covers along smoothly slice knots all bound rational homology balls. This phenomenon, however, does not characterize slice knots. In this paper, we give a new construction of non-slice knots that have the above property. The sliceness obstruction comes from computing twisted Alexander polynomials, and we introduce new techniques to simplify their calculation.

math.GT↗

arXiv2020

Rational cobordisms and integral homology

We consider the question of when a rational homology 3-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology embeds in any other element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite rank cokernels. Further consequences include a divisibility condition between the determinants of a connected sum of 2-bridge knots and any other knot in the same concordance class. Lastly, we use knot Floer homology combined with our main result to obstruct Dehn surgeries on knots from being rationally cobordant to lens spaces.

math.GT↗

arXiv2019

Embedding lens spaces in definite 4-manifolds

Every lens space has a locally flat embedding in a connected sum of 8 copies of the complex projective plane and a smooth embedding in n copies of the complex projective plane for some positive integer n. We show that there is no n such that every lens space smoothly embeds in n copies of the complex projective plane.

math.GT↗

arXiv2019

On sums of torus knots concordant to alternating knots

We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We describe some effective obstructions based on Heegaard Floer homology.

math.GT↗

arXiv2018

Pretzel links, mutation, and the slice-ribbon conjecture

Let p and q be distinct integers greater than one. We show that the 2-component pretzel link P(p,q,-p,-q) is not slice, even though it has a ribbon mutant, by using 3-fold branched covers and an obstruction based on Donaldson's diagonalization theorem. As a consequence, we prove the slice-ribbon conjecture for 4-stranded 2-component pretzel links.

math.GT↗

arXiv2017

Handle decompositions of rational balls and Casson-Gordon invariants

Given a rational homology sphere which bounds rational homology balls, we investigate the complexity of these balls as measured by the number of 1-handles in a handle decomposition. We use Casson-Gordon invariants to obtain lower bounds which also lead to lower bounds on the fusion number of ribbon knots. We use Levine-Tristram signatures to compute these bounds and produce explicit examples.

math.GT↗

arXiv2017

Knot concordance and homology sphere groups

We study two homomorphisms to the rational homology sphere group. If $ψ$ denotes the inclusion homomorphism from the integral homology sphere group, then using work of Lisca we show that the image of $ψ$ intersects trivially with the subgroup of the rational homology sphere group generated by lens spaces. As corollaries this gives a new proof that the cokernel of $ψ$ is infinitely generated, and implies that a connected sum $K$ of 2-bridge knots is concordant to a knot with determinant 1 if and only if $K$ is smoothly slice. Furthermore, if $β$ denotes the homomorphism from the knot concordance group defined by taking double branched covers of knots, we prove that the kernel of $β$ contains a $\mathbb{Z}^{\infty}$ summand by analyzing the Tristram-Levine signatures of a family of knots whose double branched covers all bound rational homology balls.

math.GT↗

arXiv2016

Embedding 3-manifolds in spin 4-manifolds

An invariant of orientable 3-manifolds is defined by taking the minimum $n$ such that a given 3-manifold embeds in the connected sum of $n$ copies of $S^2 \times S^2$, and we call this $n$ the embedding number of the 3-manifold. We give some general properties of this invariant, and make calculations for families of lens spaces and Brieskorn spheres. We show how to construct rational and integral homology spheres whose embedding numbers grow arbitrarily large, and which can be calculated exactly if we assume the 11/8-Conjecture. In a different direction we show that any simply connected 4-manifold can be split along a rational homology sphere into a positive definite piece and a negative definite piece.

math.GT↗