Constructing new spectral systems from simplicial fibrations

In this work we present an ongoing project on the development and study of new spectral systems which combine filtrations associated to Serre and Eilenberg-Moore spectral sequences of different fibrations. Our new spectral systems are part of a new module for the Kenzo system and can be useful to deduce new relations on the initial spectral sequences and to obtain information about different filtrations of the homology groups of the fiber and the base space of the fibrations.

Galois symmetries of knot spaces

We exploit the Galois symmetries of the little disks operads to show that many differentials in the Goodwillie–Weiss spectral sequences approximating the homology and homotopy of knot spaces vanish at a prime $p$ . Combined with recent results on the relationship between embedding calculus and finite-type theory, we deduce that the $(n+1)$ th Goodwillie–Weiss approximation is a $p$ -local universal Vassiliev invariant of degree $\leq n$ for every $n \leq p + 1$ .

Integration of the Kenzo System within SageMath for New Algebraic Topology Computations

This work integrates the Kenzo system within Sagemath as an interface and an optional package. Our work makes it possible to communicate both computer algebra programs and it enhances the SageMath system with new capabilities in algebraic topology, such as the computation of homotopy groups and some kind of spectral sequences, dealing in particular with simplicial objects of an infinite nature. The new interface allows computing homotopy groups that were not known before.

Counterexamples to Hochschild-Kostant-Rosenberg in characteristic p

We give counterexamples to the degeneration of the Hochschild-Kostant-Rosenberg spectral sequence in characteristic p, both in the untwisted and twisted settings. We also prove that the de Rham-HP and crystalline-TP spectral sequences need not degenerate.