Nicolas Guès

About

Research

(Pre)publications

• Linear stable ranges for homotopy groups of configuration spaces Accepted in Journal of Topology · arXiv · 2025
  Abstract

  We prove explicit linear stable ranges for the $\mathsf{FI}$-modules $\mathrm{Hom}(\pi_p \mathrm{Conf} M, \mathbb Z)$ and $\mathrm{Ext}(\pi_p \mathrm{Conf} M, \mathbb Z)$ with $\mathrm{Conf} M$ being the configuration co$\mathsf{FI}$-space of a $d$-dimensional manifold with $ d \geq 3$. The proof of this result uses a homotopy-theoretic approach to representation stability for $\mathsf{FI}$-modules. This allows us to derive representation stability results from homotopy-theoretical statements, in particular the generalized Blakers-Massey theorem. We also generalize to $\mathsf{FI}_G$-modules and orbit configuration spaces.

• Periodicity in the homology of moduli spaces of disconnected submanifolds arXiv · 2025
  Abstract

  We show that the moduli space of $n$ suitably embedded copies of a closed smooth manifold $P$ inside a closed smooth manifold $M$ satisfies cohomological periodicity over $\mathbb F_p$ when $n$ grows, with an explicit linear bound on the period and the periodicity range. This generalizes a known result about configuration spaces. We also show integral stability of the cohomology when $M$ is open, generalizing a result of Palmer and improving the slope when inverting $2$. The main input in the proof is Goodwillie and Klein's multiple disjunction lemma for embedding spaces. As a corollary we get stability and periodicity results for some classes of symmetric diffeomorphism groups of manifolds.

Other mathematical writings

• Master's thesis — "A model in groupoids for the splicing operad", written under the supervision of Paolo Salvatore.
• Research introduction note — A brief introduction to my research area for a non-specialist audience, focusing on the homotopy type of spaces of embeddings. Written in french.

Teaching

Talks

• Homotopical Methods in Representation Stability
  Topology seminar, Università Tor Vergata, Rome — April 5, 2024
• Participant in the mini working group at the 2024 meeting of the GDR in algebraic topology, Toulouse
• Participant in the working group 2023–2024 of the topology team at LAGA

Misc.

• I co-organize with Lucas Lagarde and Loth Damagui Chabi the PhD students' seminar at LAGA.
• Animations of long knots created to illustrate my Master's internship.
• A popularization talk on algebraic topology given at Lycée Thiers in September 2023.