arXiv subjects

Igor Spiridonov

Publications and source records attributed to Igor Spiridonov.

5 recordsLinked to original sources

arXiv2026

On the second homology of the genus 3 hyperelliptic Torelli group

Let $s$ be a fixed hyperelliptic involution of the closed, oriented genus $g$ surface $Σ_g$. The hyperelliptic Torelli group $\mathcal{SI}_g$ is the subgroup of the mapping class group $\mathrm{Mod}(Σ_g)$ consisting of elements that act trivially on $\mathrm{H}_1(Σ_g;\mathbb{Z})$ and commute with $s$. It is generated by Dehn twists about $s$-invariant separating curves, and its cohomological dimension is $g-1$. In this paper we study the top homology group $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$. For each pair of disjoint $s$-invariant separating curves there is a naturally associated abelian cycle in $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$; we call such cycles \emph{simple}. We show that simple abelian cycles are in bijection with orthogonal (with respect to the intersection form) splittings of $\mathrm{H}_1(Σ_3;\mathbb{Z})$ satisfying a simple algebraic condition, and prove that these abelian cycles are linearly independent in $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$.

math.GT↗

arXiv2025

Tight complexity bounds for diagram commutativity verification

A diagram $\mathcal{D} = (G, l)$ over a monoid $M$ is an oriented graph $G = (V, E)$ endowed with a labeling $l\colon E \to M$. A diagram is commutative if and only if for any two oriented paths with the same endpoints, the products in $M$ of their edge labels coincide. We propose the first asymptotically optimal algorithm for diagram commutativity verification applicable to all graph families. For graphs with $\lvert V\rvert \preceq \lvert E\rvert \preceq \lvert V\rvert^2$, which covers most practically relevant cases, our algorithm runs in $$ O\bigl(|V|\,|E|\bigr) \cdot \bigl(T_{\mathrm{equal}} + T_{\mathrm{multi}}\bigr) $$ time; here $T_{\mathrm{equal}}$ and $T_{\mathrm{multi}}$ denote the times to perform an equality check and a multiplication in $M$, respectively. We also establish new lower bounds on the numbers of equality checks and multiplications necessary for commutativity verification, which asymptotically match our algorithm's cost and thus prove its tightness.

math.CO↗

arXiv2024

On the mapping class group action on the homology of surface covers

Let $ϕ\in {\rm Mod}(Σ)$ be an arbitrary element of the mapping class group of a closed orientable surface $Σ$ of genus at least $2$. For any characteristic cover $\widetildeΣ \to Σ$ one can consider the linear subspace ${\rm H}_1^{f.o.}(\widetildeΣ, \mathbb{Q})^ϕ\subseteq {\rm H}_1(\widetildeΣ, \mathbb{Q})$ consisting of all homology classes with finite $ϕ$-orbit. We prove that $\dim {\rm H}_1^{f.o.}(\widetildeΣ, \mathbb{Q})^ϕ$ can be arbitrary large for any fixed $ϕ\in {\rm Mod}(Σ)$.

math.GT↗

arXiv2023

On linear preservers of permanental rank

Let ${\rm Mat}_n(\mathbb{F})$ denote the set of square $n\times n$ matrices over a field $\mathbb{F}$ of characteristic different from two. The permanental rank ${\rm prk}\,(A)$ of a matrix $A \in{\rm Mat}_{n}(\mathbb{F})$ is the size of the maximal square submatrix in $A$ with nonzero permanent. By $Λ^{k}$ and $Λ^{\leq k}$ we denote the subsets of matrices $A \in {\rm Mat}_{n}(\mathbb{F})$ with ${\rm prk}\,(A) = k$ and ${\rm prk}\,(A) \leq k$, respectively. In this paper for each $1 \leq k \leq n-1$ we obtain a complete characterization of linear maps $T: {\rm Mat}_{n}(\mathbb{F}) \to {\rm Mat}_{n}(\mathbb{F})$ satisfying $T(Λ^{\leq k}) = Λ^{\leq k}$ or bijective linear maps satisfying $T(Λ^{\leq k}) \subseteq Λ^{\leq k}$. Moreover, we show that if $\mathbb{F}$ is an infinite field, then $Λ^{k}$ is Zariski dense in $Λ^{\leq k}$ and apply this to describe such bijective linear maps satisfying $T(Λ^{k}) \subseteq Λ^{k}$.

math.CO↗

arXiv2022

Maximal Generalized Rank in Graphical Matrix Spaces

In this note we prove two extensions of a recent combinatorial characterization due to Li, Qiao, Wigderson, Wigderson and Zhang (arXiv:2206.04815) of the maximal dimension of bounded rank subspaces of the graphical matrix space associated with a bipartite graph. Our first result shows that the above characterization remains valid for a wide class of generalized rank functions, including e.g. the permanental rank. Our second result extends the characterization to bounded rank subspaces of the graphical alternating matrix space associated with a general graph.

math.CO↗