EPFL Topology Seminar Spring 2026

Location: Room CM 1 517 (And sometimes Zoom)

Programme

Abstracts

Kensuke Arakawa
Manifold calculus beyond space-valued functors

Manifold calculus is a homotopy-theoretic technique to study presheaves on manifolds, which decomposes them into successive approximations called polynomial approximations. First invented by Weiss to study embedding spaces, it has become an important toolset for homotopical study of manifolds.  
Like ordinary calculus, manifold calculus has two “fundamental theorems,” one which classifies polynomial presheaves, and the other that classifies homogeneous presheaves. Consistent with his goal to study embedding spaces, Weiss established these theorems for space-valued presheaves.  
From the perspective of studying manifold invariants, it is extremely natural to develop manifold calculus for presheaves with more general values, such as spectra and chain complexes. However, Weiss’s proof of the fundamental theorems relies on ad-hoc constructions on spaces, which do not seem to generalize easily. 
In this talk, I will explain that the two fundamental theorems do not depend on space-level constructions. Consequently, they extend to presheaves valued in essentially any infinity category. This talk is based on my paper “A context for manifold calculus” (arXiv:2403.03321).

Marie-Camille Delarue
Stable homology using scanning methods

Homological stability is a property that holds for various families of groups, such as the symmetric groups. We build a topological model for the monoid of the groups as a sort of category of 1-cobordisms. We can use this model to compute the group homology in a stable range by constructing a scanning map following the work of Madsen, Weiss, Galatius, Randal-Williams and others. This map allows us to express the stable homology of the groups as the homology of the infinite loop space of a certain spectrum. We will also explain how to generalize the model built for the symmetric groups to a model of the Higman–Thompson groups, which are groups of certain self-homeomorphisms of Cantor sets.

Marino Gran
Hopf formulas for cocommutative Hopf algebras

In recent years, numerous new applications of categorical Galois theory have emerged in various interesting non-abelian algebraic contexts. In particular, within semi-abelian categories, this approach has led to some new calculations of higher fundamental groups in terms of generalized commutators in categories such as that of compact groups, crossed modules, and skew braces. These categories share some structural properties with the categories of groups and of Lie algebras, and also with the category of cocommutative Hopf algebras over a field, which is also semi-abelian. This raises the natural question of whether similar homological methods can be applied to study cocommutative Hopf algebras as well.
In this talk, after reviewing some fundamental properties of semi-abelian categories and some motivating examples, I will explain that the answer to the above question is affirmative. By using the exactness properties of cocommutative Hopf algebras and the free functor universally associating a Hopf algebra with any coalgebra it is possible to establish some new Hopf-type formulae for the homology of cocommutative Hopf algebras. An important role is played by cleft extensions, namely those surjective morphisms of Hopf algebras that are split as coalgebra morphisms.
With any cleft extension, one can associate a 5-term exact sequence in homology that can be seen as a Hopf-theoretic analogue of the classical Stallings-Stammbach exact sequence in group theory. This new approach can also be applied to investigate the homology of cocommutative Hopf braces, which are interesting structures that naturally occur in the study of solutions to the so-called quantum Yang-Baxter equation. The category of cocommutative Hopf braces turns out to be both semi-abelian and monadic on the category of coalgebras, so that it is possible to investigate it from the perspective of non-abelian homological algebra.
This talk is based on a joint work with Andrea Sciandra.