An Arithmetic Topology viewpoint on Descent theory and Equivariant Categories
Miltiadis Karakikes, Sotiris Karanikolopoulos, Aristides Kontogeorgis, Dimitrios Noulas
TL;DR
This work develops an arithmetic–topology framework in which descent theory, moduli, and invariance are encoded by group actions on fundamental groups and their representations. By translating Weil’s descent cocycle into linearizations in equivariant categories, the authors unify descent for varieties and covers with categorical symmetry, and extend the paradigm to the bounded derived category $D^b(X)$. The paper proves a topological analogue of Weil’s Descent Theorem, formulates ascent data for mapping class groups, and shows that equivariant derived categories recover descended varieties, aligning arithmetic and topological invariants through monodromy representations and cohomological obstructions. These results illuminate deep structural parallels between arithmetic descent and topological ascent, providing a common toolkit for studying moduli, monodromy, and categorical quotients, with potential implications for equivariant geometry and homological mirror symmetry.
Abstract
We establish a unified group-theoretic framework bridging the arithmetic homotopy exact sequence of a variety and the Birman exact sequence of a surface. Within this framework, we reinterpret classical arithmetic notions - such as the descent of varieties and of covers - and construct their topological analogues. We formalize the parallel setting between closed subgroups of the absolute Galois group and subgroups of the Mapping Class Group of a base space and their actions on fundamental groups. This provides an analogy between arithmetic and topological invariants, allowing us to define the groups of moduli, definition, and invariance in both settings. Using this unified perspective, some purely group-theoretic proofs provide results in both settings simultaneously. Applications include a topological analogue of Weil's Descent Theorem for mapping class groups and an adaptation of Débes and Douai's cohomological obstructions regarding descent of algebraic covers to the topological setting. Finally, we elevate these results to the categorical level. We demonstrate that the classical Weil cocycle condition is equivalent to the existence of a linearization in the language of equivariant categories. Applying this perspective to the bounded derived category of coherent sheaves $\mathsf{D^b}(X)$, we show that the equivariant derived category $\mathsf{D^b}(X)^G$, under the action induced by a Weil descent datum, recovers the derived category of the descended variety.