Oisín Flynn-Connolly

Research papers

• Homotopy theory Accepted

  Higher order Massey products for algebras over algebraic operads

  Oisín Flynn-Connolly and José M. Moreno-Fernández

  Mathematische Zeitschrift, to appear

  arXivSlides

  Abstract

  We introduce higher-order Massey products for algebras over algebraic operads. This extends the work of Fernando Muro on secondary ones. We study their basic properties and behavior with respect to morphisms of algebras and operads and give some connections to formality. We prove that these higher-order operations represent the differentials in a naturally associated operadic Eilenberg–Moore spectral sequence. We also study the interplay between particular choices of higher-order Massey products and quasi-isomorphic P_∞-structures on the homology of a P-algebra. We focus on Koszul operads over a characteristic zero field and explain how our results generalize to the non-Koszul case.

• Homotopy theory Published

  Simplicial coendomorphism operads and coalgebras

  Oisín Flynn-Connolly

  Bulletin of the Australian Mathematical Society, 2026

  Journal

  Abstract

  In a recent work of Moreno-Fernandez, Wierstra and the present author [‘A recognition principle for iterated suspensions as coalgebras over the little cubes operad’, Preprint, 2022, arXiv:2210.00839], a coendomorphism operad in the category of pointed topological spaces endowed with the wedge sum was introduced. In this paper, we construct an analogue completely internal to the category of simplicial sets with the goal of defining simplicial coalgebras. As an application, we show that simplicial n-fold suspensions are coalgebras up to coherent homotopy over the Barratt–Eccles E_n-operad provided they have finitely many nondegenerate simplices.

• Homotopy theory Published

  An obstruction theory for strictly commutative algebras in positive characteristic

  Oisín Flynn-Connolly

  Algebraic & Geometric Topology, 2026

  JournalarXivSlides

  Abstract

  This is the first in a sequence of articles exploring the relationship between commutative algebras and E_∞-algebras in characteristic p and mixed characteristic. In this article we lay the groundwork by defining a new class of cohomology operations over F_p called cotriple products, generalising Massey products. We compute the secondary cohomology operations for a strictly commutative dg-algebra and the obstruction theories these induce, constructing several counterexamples to characteristic-0 behaviour, one of which answers a question of Campos, Petersen, Robert-Nicoud and Wierstra. We construct some families of higher cotriple products and comment on their behaviour. Finally, we distinguish a subclass of cotriple products that we call higher Steenrod operations and conclude with our main theorem, which says that E_∞-algebras can be rectified if and only if the higher Steenrod operations vanish coherently.

• Expository Published

  On associative and commutative differential graded algebras in positive characteristic

  Oisín Flynn-Connolly

  The Mathematical Intelligencer, 2026

  JournalPDFRelated slides
• Number theory Published

  On the divisibility of sums of Fibonacci numbers

  Oisín Flynn-Connolly

  Integers: Electronic Journal of Combinatorial Number Theory, 2025

  JournalarXivErrataTalk video

  Abstract

  We show that for infinitely many odd integers n, the sum of the first n nonzero Fibonacci numbers is divisible by n. This resolves a conjecture of Fatehizadeh and Yaqubi.

• Algebra and categorical algebra Published

  Three Schur functors related to pre-Lie algebras

  Vladimir Dotsenko and Oisín Flynn-Connolly

  Mathematical Proceedings of the Cambridge Philosophical Society, 2024

  JournalarXiv

  Abstract

  We give explicit combinatorial descriptions of three Schur functors arising in the theory of pre-Lie algebras. The first of them leads to a functorial description of the underlying vector space of the universal enveloping pre-Lie algebra of a given Lie algebra, strengthening the PBW theorem of Segal. The two other Schur functors provide functorial descriptions of the underlying vector spaces of the universal multiplicative enveloping algebra and of the module of Kähler differentials of a given pre-Lie algebra. An important consequence of such descriptions is an interpretation of the cohomology of a pre-Lie algebra with coefficients in a module as a derived functor for the category of modules over the universal multiplicative enveloping algebra.

• Algebra and categorical algebra Submitted

  Enriched coalgebras are sometimes comonadic

  Oisín Flynn-Connolly

  Preprint, 2026

  arXiv

  Abstract

  We introduce an enriched notion of a coalgebra over an operad P in a symmetric monoidal V-category C. When C is semicartesian and P is unital, we construct a V-endofunctor on C associated to P and give conditions under which it is a V-comonad with co-Eilenberg–Moore V-category isomorphic to the V-category of P-coalgebras in C. In many cases, this permits computation of V-categories of coalgebras. The key example is the category of pointed topological spaces with wedge product, enriched over topological spaces with Cartesian product, where this construction recovers the comonadic description of C_n-coalgebras of Moreno-Fernández, Wierstra and the present author. We further recover one direction of Fox's theorem.

• Algebra and categorical algebra Submitted

  On a conjecture of Dotsenko on weak nilpotence and the Engel identity

  Oisín Flynn-Connolly

  Preprint, 2026

  PDF

  Verification code (Macaulay2): 7-Engel verifier · 8-Engel verifier · 9-Engel certificate

  Abstract

  By means of AI-assisted computer algebra, we give examples of commutative but nonassociative algebras satisfying the identity t₄ = 0 such that neither the 7 nor 8-Engel identity holds. This resolves one conjecture of Dotsenko in the negative, and confirms a second. Conversely, we show that t₄ = 0 implies the 9-Engel identity. Together these show that 9 is the least integer m for which t₄ = 0 implies the m-Engel identity.

• Theoretical computer science Preprint

  Central limits via dilated categories

  Henning Basold, Oisín Flynn-Connolly, Chase Ford and Hao Wang

  Preprint, 2026

  arXivSlides

  Summary

  Our main result is a categorical version of the central limit theorem. Additionally, we introduce a new category to act as a base of enrichment and prove a categorical version of the Banach fixed point theorem.

• Homotopy theory Draft under revision

  A p-adic de Rham complex

  Oisín Flynn-Connolly

  arXiv manuscript, 2025

  arXiv

  The linked arXiv version has a known proof issue; a revised version is in preparation.

• Homotopy theory Preprint

  A recognition principle for iterated suspensions as coalgebras over the little cubes operad

  Oisín Flynn-Connolly, José M. Moreno-Fernández and Felix Wierstra

  Preprint, 2022

  arXivPoster

  Abstract

  Our main result is a recognition principle for iterated suspensions as coalgebras over the little disks operads. Given a topological operad, we construct a comonad in pointed topological spaces endowed with the wedge product. We then prove an approximation theorem that shows that the comonad associated to the little n-cubes operad is weakly equivalent to the comonad ΣⁿΩⁿ arising from the suspension-loop space adjunction. Finally, our recognition theorem states that every little n-cubes coalgebra is homotopy equivalent to an n-fold suspension. These results are the Eckmann–Hilton dual of May's foundational results on iterated loop spaces.