arXiv subjects

Martina Iannella

Publications and source records attributed to Martina Iannella.

5 recordsLinked to original sources

arXiv2026

Borel classification of simplicial complexes and non-compact $2$- and $3$-manifolds

We generalize the Stone space of ultrafilters on Boolean algebras and prove a generalization of Stone duality which is applicable to locally compact Polish spaces. Using this, we obtain complete invariants for simplicial complexes up to PL-homeomorphism and for non-compact $2$- and $3$-manifolds up to homeomorphism. We prove that the homeomorphism relation on non-compact $2$-manifolds without boundary, the homeomorphism relation on non-compact $3$-manifolds with or without boundary, the homeomorphism relation on open subsets of $\mathbb{R}^2$ and $\mathbb{R}^3$, and conjugacy of Cantor sets in $\mathbb{R}^3$ are classifiable by countable structures. Together with known lower bounds, this implies that these relations are Borel bireducible with isomorphism of countable graphs. We also show that PL-homeomorphism of Heine-Borel simplicial complexes and PL-homeomorphism of PL $n$-manifolds, for every $n$, are classifiable by countable structures.

math.LO↗

arXiv2026

Alexander's conjecture for infinite simplicial complexes

Alexander's conjecture states that for every two finite triangulations of the same topological space, if they have a common subdivision, then they have a common stellar subdivision. We generalize the recent result of Adiprasito and Pak, who resolved Alexander's conjecture for finite simplicial complexes, to infinite simplicial complexes.

math.GT↗

arXiv2024

Descriptive properties of I2-embeddings

We contribute to the study of generalizations of the Perfect Set Property and the Baire Property to subsets of spaces of higher cardinalities, like the power set $P(λ)$ of a singular cardinal $λ$ of countable cofinality or products $\prod_{i<ω}λ_i$ for a strictly increasing sequence $\langleλ_i ~ \vert ~ i<ω\rangle$ of cardinals. We consider the question under which large cardinal hypotheses classes of definable subsets of these spaces possess such regularity properties, focusing on rank-into-rank axioms and classes of sets definable by $Σ_1$-formulas with parameters from various collections of sets. We prove that $ω$-many measurable cardinals, while sufficient to prove the Perfect Set Property of all $Σ_1$-definable sets with parameters in $V_λ\cup\{V_λ\}$, are not enough to prove it if there is a cofinal sequence in $λ$ in the parameters. For this conclusion, the existence of an I2-embedding is enough, but there are parameters in $V_{λ+1}$ for which I2 is still not enough. The situation is similar for the Baire Property: under I2 all sets that are $Σ_1$-definable using elements of $V_λ$ and a cofinal sequence as parameters have the Baire property, but I2 is not enough for some parameter in $V_{λ+1}$. Finally, the existence of an I0-embedding implies that all sets that are $Σ^1_n$-definable with parameters in $V_{λ+1}$ have the Baire property.

math.LO↗

arXiv2023

Piecewise convex embeddability on linear orders

Given a nonempty set $\mathcal{L}$ of linear orders, we say that the linear order $L$ is $\mathcal{L}$-convex embeddable into the linear order $L'$ if it is possible to partition $L$ into convex sets indexed by some element of $\mathcal{L}$ which are isomorphic to convex subsets of $L'$ ordered in the same way. This notion generalizes convex embeddability and (finite) piecewise convex embeddability (both studied in arXiv:2309.09910), which are the special cases $\mathcal{L} = \{\mathbf{1}\}$ and $\mathcal{L} = \mathsf{Fin}$. We focus mainly on the behavior of these relations on the set of countable linear orders, first characterizing when they are transitive, and hence a quasi-order. We then study these quasi-orders from a combinatorial point of view, and analyze their complexity with respect to Borel reducibility. Finally, we extend our analysis to uncountable linear orders.

math.LO↗

arXiv2023

Convex Embeddability and Knot Theory

We consider countable linear orders and study the quasi-order of convex embeddability and its induced equivalence relation. We obtain both combinatorial and descriptive set-theoretic results, and further extend our research to the case of circular orders. These results are then applied to the study of arcs and knots, establishing combinatorial properties and lower bounds (in terms of Borel reducibility) for the complexity of some natural relations between these geometrical objects.

math.LO↗