arXiv subjects

Najib Idrissi

Publications and source records attributed to Najib Idrissi.

13 recordsLinked to original sources

arXiv2026

Odd-primary torsion in the homology of unordered configurations on the torus

The main object of study of this paper is $B_k(Σ_1)$, the unordered configuration space of $k$ points in the torus $Σ_1$. First, we produce a marked-point transfer argument which, combined with Chen--Zhang's theorem, establishes that $H_*(B_k (Σ_1);\mathbb{Z})$ has no $p$-torsion for $k\leq 2p-1$. Secondly, at the threshold $k=2p$, we prove that $H_{2p-2}(B_{2p}(Σ_1);\mathbb{Z})$ has $p$-torsion if and only if $p\geq 5$. For $p\geq5$, the class is the image under puncture filling of a generator of the Bianchi--Stavrou torsion group $\mathbb{Z}/p$ of the once-punctured torus; for $p=3$, Napolitano's calculation shows that this punctured class dies after filling. Finally, we also prove that $H_{2p}(B_{2p}(Σ_1);\mathbb{Z})$ and $H_{2p+1}(B_{2p}(Σ_1);\mathbb{Z})$ have no $p$-torsion for every odd $p$.

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arXiv2026

Homology of configuration spaces in positive characteristic via point-set constructions

The first goal of this paper is to provide concrete chain complexes computing the homology of (unordered) configuration spaces of manifolds in positive characteristic, lifting a theorem by Knudsen to the model category level. We make them fully explicit and provide a computer program to compute their homology. Our methods also allow us to construct several new spectral sequences converging to these homology groups. Finally, we conjecture that this equivalence of chain complexes can be promoted to an equivalence of \emph{twisted} $\EE_\infty$-coalgebras in right $\EE_d$-modules, and we explain how this conjecture would imply the homotopy invariance of the $\EE_d$-homotopy type of configuration spaces in positive characteristic via new ``twist'' and ``detwist'' functors.

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arXiv2025

Plethysm for characters of relative operads and PROPs

We investigate the relationship between symmetric functions and the representation theory of operads, relative operads, and props. We extend the classical character map for symmetric sequences to relative bisymmetric sequences and symmetric bimodules. We introduce new operations on symmetric functions, the relative plethysm and the (connected) box product, which model via the character map the composition product of relative operads and the box product of prop(erad)s. As applications, we include the computation of characters for stable twisted cohomology of automorphism groups of free groups and the Albanese cohomology of the IA-automorphism group.

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arXiv2025

A model for framed configuration spaces of points

We study configuration spaces of framed points on oriented closed smooth manifolds. Such configuration spaces admit natural actions of the framed little discs operads, that play an important role in the study of embedding spaces of manifolds and in factorization homology. We construct real combinatorial models for these operadic modules, for orientable closed smooth manifolds.

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arXiv2024

Homotopy Prefactorization Algebras

We apply the theory of operadic Koszul duality to provide a cofibrant resolution of the colored operad whose algebras are prefactorization algebras on a fixed space M. his allows us to describe a notion of prefactorization algebra up to homotopy as well as morphisms up to homotopy between such objects. We make explicit these notions for several special M, such as certain finite topological spaces, or the real line.

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arXiv2023

Non-formality of Voronov's Swiss-Cheese operads

The Swiss-Cheese operads, which encode actions of algebras over the little $n$-cubes operad on algebras over the little $(n-1)$-cubes operad, comes in several variants. We prove that the variant in which open operations must have at least one open input is not formal in characteristic zero. This is slightly stronger than earlier results of Livernet and Willwacher. The obstruction to formality that we find lies in arity $(2, 2^n)$, rather than $(2, 0)$ (Livernet) or $(4, 0)$ (Willwacher).

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arXiv2023

Configuration Spaces of Manifolds with Boundary

We study ordered configuration spaces of compact manifolds with boundary. We show that for a large class of such manifolds, the real homotopy type of the configuration spaces only depends on the real homotopy type of the pair consisting of the manifold and its boundary. We moreover describe explicit real models of these configuration spaces using three different approaches. We do this by adapting previous constructions for configuration spaces of closed manifolds which relied on Kontsevich's proof of the formality of the little disks operads. We also prove that our models are compatible with the richer structure of configuration spaces, respectively a module over the Swiss-Cheese operad, a module over the associative algebra of configurations in a collar around the boundary of the manifold, and a module over the little disks operad.

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arXiv2022

Curved Koszul duality of algebras over unital versions of binary operads

We develop a curved Koszul duality theory for algebras presented by quadratic-linear-constant relations over unital versions of binary quadratic operads. As an application, we study Poisson $n$-algebras given by polynomial functions on a standard shifted symplectic space. We compute explicit resolutions of these algebras using curved Koszul duality. We use these resolutions to compute derived enveloping algebras and factorization homology on parallelized simply connected closed manifolds with coefficients in these Poisson $n$-algebras.

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arXiv2021

Boardman-Vogt resolutions and bar/cobar constructions of (co)operadic (co)bimodules

We develop the combinatorics of leveled trees in order to construct explicit resolutions of (co)operads and (co)operadic (co)bimodules. We build explicit cofibrant resolutions of operads and operadic bimodules in spectra analogous to the ordinary Boardman--Vogt resolutions and we express them as cobar constructions of indecomposable elements. Dually, in the context of CDGAs, we perform similar constructions, and we obtain fibrant resolutions of Hopf cooperads and Hopf cooperadic cobimodules. We also express them as bar constructions of primitive elements.

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arXiv2020

Formality of a higher-codimensional Swiss-Cheese operad

We study bicolored configurations of points in the Euclidean $n$-space that are constrained to remain either inside or outside a fixed Euclidean $m$-subspace, with $n - m \ge 2$. We define a higher-codimensional variant of the Swiss-Cheese operad, called the complementarily constrained disks operad $\mathsf{CD}_{mn}$, associated to such configurations. The operad $\mathsf{CD}_{mn}$ is weakly equivalent to the operad of locally constant factorization algebras on the stratified space $\{\mathbb{R}^{m} \subset \mathbb{R}^{n}\}$. We prove that this operad is formal over $\mathbb{R}$.

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arXiv2020

Configuration Spaces of Surfaces

We compute small rational models for configuration spaces of points on oriented surfaces, as right modules over the framed little disks operad. We do this by splitting these surfaces in unions of several handles. We first describe rational models for the configuration spaces of these handles as algebras in the category of right modules over the framed little disks operad. We then express the configuration spaces of the surface as an "iterated Hochschild complex" of these algebras with coefficients in the module given by configurations in a sphere with holes. Physically, our results may be interpreted as saying that the partition function of the Poisson-$σ$-model on closed surfaces has no quantum corrections, i.e., no terms coming from Feynman diagrams of positive loop order.

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arXiv2018

The Lambrechts-Stanley Model of Configuration Spaces

We prove the validity over $\mathbb{R}$ of a commutative differential graded algebra model of configuration spaces for simply connected closed smooth manifolds, answering a conjecture of Lambrechts--Stanley. We get as a result that the real homotopy type of such configuration spaces only depends on the real homotopy type of the manifold. We moreover prove, if the dimension of the manifold is at least $4$, that our model is compatible with the action of the Fulton--MacPherson operad (weakly equivalent to the little disks operad) when the manifold is framed. We use this more precise result to get a complex computing factorization homology of framed manifolds. Our proofs use the same ideas as Kontsevich's proof of the formality of the little disks operads.

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arXiv2017

Swiss-Cheese operad and Drinfeld center

We build a model in groupoids for the Swiss-Cheese operad, based on parenthesized permutations and braids, and we relate algebras over this model to the classical description of algebras over the homology of the Swiss-Cheese operad. We extend our model to a rational model for the Swiss-Cheese operad, and we compare it to the model that we would get if the operad Swiss-Cheese were formal.

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