arXiv subjects

Tudur Lewis

Publications and source records attributed to Tudur Lewis.

3 recordsLinked to original sources

arXiv2025

Representation stability for the first homology of congruence subgroups

We study sequences of modular representations of the symplectic and special linear groups over finite fields arising from the first homology of congruence subgroups of mapping class groups and automorphism groups of free groups, as well as from the module of coinvariants for the abelianization of the Torelli group. In each case, we determine the composition factors and their multiplicities, and establish periodic representation stability in the sense of Church--Farb. We apply our results to study flat line bundles over the moduli space of curves with level 2 structure arising from spin structures on the underlying surface.

math.GT↗

arXiv2025

Cyclic Nielsen realization for del Pezzo surfaces

The cyclic Nielsen realization problem for a closed, oriented manifold asks whether any mapping class of finite order can be represented by a homeomorphism of the same order. In this article, we resolve the smooth, metric, and complex cyclic Nielsen realization problem for certain "irreducible" mapping classes on the family of smooth 4-manifolds underlying del Pezzo surfaces. Both positive and negative examples of realizability are provided in various settings. Our techniques are varied, synthesizing results from reflection group theory and 4-manifold topology.

math.GT↗

arXiv2022

A surgery approach to abelian quotients of the level 2 congruence group and the Torelli group

We provide algorithms for computing the Rochlin invariants of mod 2 homology spheres and mapping tori. This provides a unified framework for studying two families of maps: the Birman-Craggs maps of the Torelli group, and Sato's maps of the level 2 congruence subgroup of the mapping class group. Our framework gives new, elementary proofs that both families of maps are homomorphisms, gives an explicit method for evaluating these maps on Dehn twists, and relates the two families when restricted to the Torelli group. It also gives a relation between an extension of the Birman-Craggs maps to the level 2 congruence subgroup, and Meyer's signature cocycle.

math.GT↗