Trent Lucas

Papers and preprints

1. Isotopy versus equivariant isotopy in dimensions three and higher. Preprint. (pdf, arxiv)
   Abstract

   Given a finite group action on a smooth manifold, we study the following question: if two equivariant diffeomorphisms are isotopic, must they be equivariantly isotopic? Birman-Hilden and Maclachlan-Harvey proved the answer is "yes" for most surfaces. By contrast, we give a general criterion in higher dimensions under which there are many equivariant diffeomorphisms which are isotopic but not equivariantly isotopic. Examples satisfying this criterion include branched covers of split links and "stabilized" branched covers. We prove the result by constructing an invariant valued in the homology of a certain infinite cover of the manifold. We give applications to outer automorphism groups of free products and to group actions on manifolds which fiber over the circle.

2. Birman-Hilden theory for 3-manifolds. Adv. Math. 468 (2025) 110204. (pdf, arxiv)
   Abstract

   Given a branched cover of manifolds, one can lift homeomorphisms along the cover to obtain a (virtual) homomorphism between mapping class groups. Following a question of Margalit-Winarski, we study the injectivity of this lifting map in the case of 3-manifolds. We show that in contrast to the case of surfaces, the lifting map is generally not injective for most regular branched covers of 3-manifolds. This includes the double cover of S³ branched over the unlink, which generalizes the hyperelliptic branched cover of S². In this case, we find a finite normal generating set for the kernel of the lifting map.

3. Veech fibrations. With Sam Freedman, submitted. (pdf, arxiv)
   Abstract

   We investigate complex surfaces that fiber over Teichmüller curves where the generic fiber is a Veech surface. When the fiber has genus one, these surfaces are elliptic fibrations; for higher genus fibers, they are typically minimal surfaces of general type. We compute the topological and complex-geometric invariants of these surfaces via the monodromy action on the mod-m homology of the fiber. We get exact values of the invariants for all known algebraically primitive Teichmüller curves.

4. Homological representations of low genus mapping class groups. J. Algebra 647 (2024) 533-566. (pdf, arxiv)
   Abstract

   Given a finite group G acting on a surface S, the centralizer of G in the mapping class group Mod(S) has a natural representation given by its action on the homology H_1(S;Q). We consider the question of whether this representation has arithmetic image. Several authors have given positive and negative answers to this question. We give a complete answer when S has genus at most 3.

Notes and other writing

• Hyperbolic structures on surfaces. Undergraduate thesis. (pdf)

Recent and upcoming travel

• June 9-20: Discrete Groups in Topology and Algebraic Geometry (Notre Dame)
• May 12-16: Links in Dimensions 3 and 4 (ICERM)
• March 6-8: Spring Topology and Dynamics Conference (Christopher Newport University)

Here is a list of some courses I have taught recently.

Brown University
• Instructor, MATH 520 Linear Algebra, Spring 2025
• TA, CEMA 922 Set Theory (Brown Pre-College Program), Summer 2024
• Instructor, MATH 50 Analytic Geometry and Calculus, Fall 2023
• Instructor, MATH 180 Multivariable Calculus, Spring 2023
• TA, MATH 180 Multivariable Calculus, Fall 2022
• TA, MATH 90 Calculus I, Spring 2022
• TA, MATH 90 Calculus I, Fall 2021
University of Virginia
• TA, MATH 3310 Basic Real Analysis, Spring 2020
• TA, MATH 3310 Basic Real Analysis, Fall 2019