arXiv subjects

Eric Samperton

Publications and source records attributed to Eric Samperton.

16 recordsLinked to original sources

arXiv2026

Topological perspectives on the vanishing of some Bogomolov multipliers

Since the 1980s, the Bogomolov multiplier of a finite group has been known to obstruct rationality in complex algebraic geometry, and more recently it is understood to be responsible for any torsion in the oriented and stable unitary 2-dimensional $G$-equivariant bordism groups $\Omega_2^{SO,G}$ and $\Omega_2^{U,G}$. In this note, as a small step toward building a bridge between these two far-flung roles, we discuss the vanishing of Bogomolov multipliers of two specific families of finite groups. First, we revisit Kunyavski\u{i}'s result that the Bogomolov multipliers of all finite simple groups vanish, taking inspiration from the low-dimensional topological interpretation of the Ore conjecture. Second, in lieu of arguments in complex birational geometry (such as the hard direction of the Chevalley-Shephard-Todd theorem), we combine cut-and-paste combinatorial-topological techniques with elementary calculations of Ihara-Yokonuma to show that all finite Coxeter groups have vanishing Bogomolov multiplier.

math.GR↗

arXiv2026

Obstruction theory and the complexity of counting group homomorphisms

Fix a finite group $G$. We study the computational complexity of counting problems of the following flavor: given a group $\Gamma$, count the number of homomorphisms $\Gamma \to G$. Our first result establishes that this problem is $\#\mathsf{P}$-hard whenever $G$ is a non-abelian group and $\Gamma$ is provided via a finite presentation. We give several improvements showing that this hardness conclusion continues to hold for restricted $\Gamma$ satisfying various promises. Our second result shows that if $G$ is class 2 nilpotent and $\Gamma = \pi_1(M^3)$ for some input 3-manifold triangulation $M^3$ with $|H^2(M,Z(G)|$ bounded above, then there is a polynomial time algorithm to compute the number of homomorphisms from $\Gamma$ to $G$. This algorithm is explained in part by the fact that 3-manifolds are close enough to being Eilenberg-MacLane spaces for us to solve the necessary group cohomological obstruction problems efficiently using the given triangulation. A similar polynomial time algorithm for counting maps to finite, class 2 nilpotent $G$ exists when $\Gamma$ is itself a finite group encoded via a multiplication table, provided that $|H^2(\Gamma,Z(G))|$ is similarly bounded from above.

math.GR↗

arXiv2025

On the hardness of approximating minimum distances of quantum codes

The problem of computing distances of error-correcting codes is fundamental in both the classical and quantum settings. While hardness for the classical version of these problems has been known for some time (in both the exact and approximate settings), it was only recently that Kapshikar and Kundu showed these problems are also hard in the quantum setting. As our first main result, we reprove this using arguably simpler arguments based on hypergraph product codes. In particular, we get a direct reduction to CSS codes, the most commonly used type of quantum code, from the minimum distance problem for classical linear codes. Our second set of results considers the distance of a graph state, which is a key parameter for quantum codes obtained via the codeword stabilized formalism. We show that it is NP-hard to compute/approximate the distance of a graph state when the adjacency matrix of the graph is the input. In fact, we show this is true even if we only consider X-type errors of a graph state. Our techniques moreover imply an interesting classical consequence: the hardness of computing or approximating the distance of classical codes with rate equal to 1/2. One of the main motivations of the present work is a question raised by Kapshikar and Kundu concerning the NP-hardness of approximation when there is an additive error proportional to a quantum code's length. We show that no such hardness can hold for hypergraph product codes. These observations suggest the possibility of a new kind of square root barrier.

quant-ph↗

arXiv2025

An elementary proof that linking problems are hard

We give a new, elementary proof of what we believe is the simplest known example of a ``natural'' problem in computational 3-dimensional topology that is $\mathsf{NP}$-hard -- namely, the \emph{Trivial Sublink Problem}: given a diagram $L$ of a link in $S^3$ and a positive integer $k$, decide if $L$ contains a $k$ component sublink that is trivial. This problem was previously shown to be $\mathsf{NP}$-hard in independent works of Koenig-Tsvietkova and de Mesmay-Rieck-Sedgwick-Tancer, both of which used reductions from $\mathsf{3SAT}$. The reduction we describe instead starts with the Independent Set Problem, and allows us to avoid the use of Brunnian links such as the Borromean rings. On the technical level, this entails a new conceptual insight: the Trivial Sublink Problem is hard entirely due to mod 2 pairwise linking, with no need for integral or higher order linking. On the pedagogical level, the reduction we describe is entirely elementary, and thus suitable for introducing undergraduates and non-experts to complexity-theoretic low-dimensional topology. To drive this point home, in this work we assume no familiarity with low-dimensional topology, and -- other than Reidemeister's Theorem and Karp's result that the Clique Problem is $\mathsf{NP}$-hard -- we provide more-or-less complete definitions and proofs. We have also constructed a web app that accompanies this work and allows a user to visualize the new reduction interactively.

cs.CC↗

arXiv2025

Towards a complexity-theoretic dichotomy for TQFT invariants

We show that for any fixed $(2+1)$-dimensional TQFT over $\mathbb{C}$ of either Turaev-Viro-Barrett-Westbury or Reshetikhin-Turaev type, the problem of (exactly) computing its invariants on closed 3-manifolds is either solvable in polynomial time, or else it is $\#\mathsf{P}$-hard to (exactly) contract certain tensors that are built from the TQFT's fusion category. Our proof is an application of a dichotomy result of Cai and Chen [J. ACM, 2017] concerning weighted constraint satisfaction problems over $\mathbb{C}$. We leave for future work the issue of reinterpreting the conditions of Cai and Chen that distinguish between the two cases (i.e. $\#\mathsf{P}$-hard tensor contractions vs. polynomial time invariants) in terms of fusion categories. We expect that with more effort, our reduction can be improved so that one gets a dichotomy directly for TQFTs' invariants of 3-manifolds rather than more general tensors built from the TQFT's fusion category.

math.QA↗

arXiv2023

An algorithm for Tambara-Yamagami quantum invariants of 3-manifolds, parameterized by the first Betti number

Quantum topology provides various frameworks for defining and computing invariants of manifolds inspired by quantum theory. One such framework of substantial interest in both mathematics and physics is the Turaev-Viro-Barrett-Westbury state sum construction, which uses the data of a spherical fusion category to define topological invariants of triangulated 3-manifolds via tensor network contractions. In this work we analyze the computational complexity of state sum invariants of 3-manifolds derived from Tambara-Yamagami categories. While these categories are the simplest source of state sum invariants beyond finite abelian groups (whose invariants can be computed in polynomial time) their computational complexities are yet to be fully understood. We first establish that the invariants arising from even the smallest Tambara-Yamagami categories are #P-hard to compute, so that one expects the same to be true of the whole family. Our main result is then the existence of a fixed parameter tractable algorithm to compute these 3-manifold invariants, where the parameter is the first Betti number of the 3-manifold with Z/2Z coefficients. Contrary to other domains of computational topology, such as graphs on surfaces, very few hard problems in 3-manifold topology are known to admit FPT algorithms with a topological parameter. However, such algorithms are of particular interest as their complexity depends only polynomially on the combinatorial representation of the input, regardless of size or combinatorial width. Additionally, in the case of Betti numbers, the parameter itself is computable in polynomial time. Thus while one generally expects quantum invariants to be hard to compute classically, our results suggest that the hardness of computing state sum invariants from Tambara-Yamagami categories arises from classical 3-manifold topology rather than the quantum nature of the algebraic input.

cs.CG↗

arXiv2022

Topological quantum computation is hyperbolic

We show that a topological quantum computer based on the evaluation of a Witten-Reshetikhin-Turaev TQFT invariant of knots can always be arranged so that the knot diagrams with which one computes are diagrams of hyperbolic knots. The diagrams can even be arranged to have additional nice properties, such as being alternating with minimal crossing number. Moreover, the reduction is polynomially uniform in the self-braiding exponent of the coloring object. Various complexity-theoretic hardness results regarding the calculation of quantum invariants of knots follow as corollaries. In particular, we argue that the hyperbolic geometry of knots is unlikely to be useful for topological quantum computation.

quant-ph↗

arXiv2021

Oriented and unitary equivariant bordism of surfaces

Fix a finite group $G$. We study $\Omega^{SO,G}_2$ and $\Omega^{U,G}_2$, the unitary and oriented bordism groups of smooth $G$-equivariant compact surfaces, respectively, and we calculate them explicitly. Their ranks are determined by the possible representations around fixed points, while their torsion subgroups are isomorphic to the direct sum of the Bogomolov multipliers of the Weyl groups of representatives of conjugacy classes of all subgroups of $G$. We present an alternative proof of the fact that surfaces with free actions which induce non-trivial elements in the Bogomolov multiplier of the group cannot equivariantly bound. This result permits us to show that the 2-dimensional SK-groups (Schneiden und Kleben, or ``cut and paste") of the classifying spaces of a finite group can be understood in terms of the bordism group of free equivariant surfaces modulo the ones that bound arbitrary actions.

math.AT↗

arXiv2021

Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds

We provide the first known example of a finite group action on an oriented surface $T$ that is free, orientation-preserving, and does not extend to an arbitrary (in particular, possibly non-free) orientation-preserving action on any compact oriented 3-manifold $N$ with boundary $\partial N = T$. This implies a negative solution to a conjecture of Dom\'inguez and Segovia, as well as Uribe's evenness conjecture for equivariant unitary bordism groups. We more generally provide sufficient conditions that imply infinitely many such group actions on surfaces exist. Intriguingly, any group with such a non-extending action is also a counterexample to the Noether problem over the complex numbers $\mathbb{C}$. In forthcoming work with Segovia we give a complete homological characterization of those finite groups admitting such a non-extending action, as well as more examples and non-examples. We do not address here the analogous question for non-orientation-preserving actions.

math.GT↗

arXiv2019

Coloring invariants of knots and links are often intractable

Let $G$ be a nonabelian, simple group with a nontrivial conjugacy class $C \subseteq G$. Let $K$ be a diagram of an oriented knot in $S^3$, thought of as computational input. We show that for each such $G$ and $C$, the problem of counting homomorphisms $\pi_1(S^3\setminus K) \to G$ that send meridians of $K$ to $C$ is almost parsimoniously $\mathsf{\#P}$-complete. This work is a sequel to a previous result by the authors that counting homomorphisms from fundamental groups of integer homology 3-spheres to $G$ is almost parsimoniously $\mathsf{\#P}$-complete. Where we previously used mapping class groups actions on closed, unmarked surfaces, we now use braid group actions.

math.GT↗

arXiv2018

Haah codes on general three manifolds

Haah codes represent a singularly interesting gapped Hamiltonian schema that has resisted a natural generalization, although recent work shows that the closely related type I fracton models are more commonplace. These type I siblings of Haah codes are better understood, and a generalized topological quantum field theory framework has been proposed. Following the same conceptual framework, we outline a program to generalize Haah codes to all 3-manifolds using Hastings' LR stabilizer codes for finite groups.

quant-ph↗

arXiv2018

Computational Complexity of Enumerative 3-Manifold Invariants

Fix a finite group $G$. We analyze the computational complexity of the problem of counting homomorphisms $\pi_1(X) \to G$, where $X$ is a topological space treated as computational input. We are especially interested in requiring $G$ to be a fixed, finite, nonabelian, simple group. We then consider two cases: when the input $X=M$ is a closed, triangulated 3-manifold, and when $X=S^3 \setminus K$ is the complement of a knot (presented as a diagram) in $S^3$. We prove complexity theoretic hardness results in both settings. When $M$ is closed, we show that counting homomorphisms $\pi_1(M) \to G$ (up to automorphisms of $G$) is $\#\mathsf{P}$-complete via parsimonious Levin reduction---the strictest type of polynomial-time reduction. This remains true even if we require $M$ to be an integer homology 3-sphere. We prove an analogous result in the case that $X=S^3 \setminus K$ is the complement of a knot. Both proofs proceed by studying the action of the pointed mapping class group $\mathrm{MCG}_*(\Sigma)$ on the set of homomorphisms $\{\pi_1(\Sigma) \to G\}$ for an appropriate surface $\Sigma$. In the case where $X=M$ is closed, we take $\Sigma$ to be a closed surface with large genus. When $X=S^3 \setminus K$ is a knot complement, we take $\Sigma$ to be a disk with many punctures. Our constructions exhibit classical computational universality for a combinatorial topological quantum field theory associated to $G$. Our "topological classical computing" theorems are analogs of the famous results of Freedman, Larsen and Wang establishing the quantum universality of topological quantum computing with the Jones polynomial at a root of unity. Instead of using quantum circuits, we develop a circuit model for classical reversible computing that is equivariant with respect to a symmetry of the computational alphabet.

math.GT↗

arXiv2017

Schur-type invariants of branched G-covers of surfaces

Fix a finite group $G$ and a conjugacy invariant subset $C\subseteq G$. Let $\Sigma$ be an oriented surface, possibly with punctures. We consider the question of when two homomorphisms $\pi_1(\Sigma) \to G$ taking punctures into $C$ are equivalent up to an orientation preserving diffeomorphism of $\Sigma$. We provide an answer to this question in a stable range, meaning that $\Sigma$ has enough genus and enough punctures of every conjugacy type in $C$. If $C$ generates $G$, then we can assume $\Sigma$ has genus 0 (or any other constant). The main tool is a classifying space for (framed) $C$-branched $G$-covers, and related homology classes we call branched Schur invariants, since they take values in a torsor over a quotient of the Schur multiplier $H_2(G)$. We conclude with a brief discussion of applications to $(2+1)$-dimensional $G$-equivariant TQFT and symmetry-enriched topological phases.

math.GT↗

arXiv2017

Computational complexity and 3-manifolds and zombies

We show the problem of counting homomorphisms from the fundamental group of a homology $3$-sphere $M$ to a finite, non-abelian simple group $G$ is #P-complete, in the case that $G$ is fixed and $M$ is the computational input. Similarly, deciding if there is a non-trivial homomorphism is NP-complete. In both reductions, we can guarantee that every non-trivial homomorphism is a surjection. As a corollary, for any fixed integer $m \ge 5$, it is NP-complete to decide whether $M$ admits a connected $m$-sheeted covering. Our construction is inspired by universality results in topological quantum computation. Given a classical reversible circuit $C$, we construct $M$ so that evaluations of $C$ with certain initialization and finalization conditions correspond to homomorphisms $\pi_1(M) \to G$. An intermediate state of $C$ likewise corresponds to a homomorphism $\pi_1(\Sigma_g) \to G$, where $\Sigma_g$ is a pointed Heegaard surface of $M$ of genus $g$. We analyze the action on these homomorphisms by the pointed mapping class group $\text{MCG}_*(\Sigma_g)$ and its Torelli subgroup $\text{Tor}_*(\Sigma_g)$. By results of Dunfield-Thurston, the action of $\text{MCG}_*(\Sigma_g)$ is as large as possible when $g$ is sufficiently large; we can pass to the Torelli group using the congruence subgroup property of $\text{Sp}(2g,\mathbb{Z})$. Our results can be interpreted as a sharp classical universality property of an associated combinatorial $(2+1)$-dimensional TQFT.

math.GT↗

arXiv2015

Spaces of invariant circular orders of groups

Motivated by well known results in low-dimensional topology, we introduce and study a topology on the set CO(G) of all left-invariant circular orders on a fixed countable and discrete group G. CO(G) contains as a closed subspace LO(G), the space of all left-invariant linear orders of G, as first topologized by Sikora. We use the compactness of these spaces to show the sets of non-linearly and non-circularly orderable finitely presented groups are recursively enumerable. We describe the action of Aut(G) on CO(G) and relate it to results of Koberda regarding the action on LO(G). We then study two families of circularly orderable groups: finitely generated abelian groups, and free products of circularly orderable groups. For finitely generated abelian groups A, we use a classification of elements of CO(A) to describe the homeomorphism type of the space CO(A), and to show that Aut(A) acts faithfully on the subspace of circular orders which are not linear. We define and characterize Archimedean circular orders, in analogy with linear Archimedean orders. We describe explicit examples of circular orders on free products of circularly orderable groups, and prove a result about the abundance of orders on free products. Whenever possible, we prove and interpret our results from a dynamical perspective.

math.GR↗

arXiv2014

On laminar groups, Tits alternatives, and convergence group actions on $S^2$

Following previous work of the second author, we establish more properties of groups of circle homeomorphisms which admit invariant laminations. In this paper, we focus on a certain type of such groups-so-called pseudo-fibered groups, and show that many 3-manifold groups are examples of pseudo-fibered groups. We then prove that torsion-free pseudo-fibered groups satisfy a Tits alternative. We conclude by proving that a purely hyperbolic pseudo-fibered group acts on the 2-sphere as a convergence group. This leads to an interesting question if there are examples of pseudo-fibered groups other than 3-manifold groups.

math.GT↗