arXiv subjects

Chloe Lewis

Publications and source records attributed to Chloe Lewis.

5 recordsLinked to original sources

arXiv2026

Parametrized $\underline{\mathbb{F}}_2$-Cohomology of $B_{C_2}O(1)$

We compute the parametrized (or twisted) ordinary cohomology of the classifying space for real $C_2$-line bundles, $B_{C_2}O(1)$ with coefficients in the constant Mackey functor $\underline{\mathbb{F}}_2$. Parametrized cohomology refines $RO(G)$-graded Bredon cohomology by assembling equivariant cohomology for all local coefficients into a single graded ring. For this reason, our work also encodes a computation of the $RO(C_2)$-graded cohomology of all Thom spaces of real $C_2$-vector bundles over $B_{C_2}O(1)$. Along the way, we prove many general results that can be applied to computations of parametrized cohomology over general bases $B$. In particular, we introduce a collection of characteristic classes, give a definition of orientation for non-homogeneous bundles, import equivariant Steenrod operations to this context, and give general results relating to base change.

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arXiv2025

Characterizing Transfer Systems for Non-Abelian Groups

For a finite group $G$, the notion of a $G$-transfer system provides homotopy theorists with a combinatorial way to study equivariant objects. In this paper, we focus on the properties of transfer systems for non-abelian groups. We explicitly describe the width of all dihedral groups, quaternion groups, and dicyclic groups. For a given $G$, the set of all $G$-transfer systems forms a poset lattice under inclusion; these are a useful resource to homotopical combinatorialists for detecting patterns and checking conjectures. We expand the suite of known transfer system lattices for non-abelian groups including those which are dihedral, dicyclic, Frobenius, and alternating.

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arXiv2025

Combinatorics of factorization systems on lattices

We initiate the combinatorial study of factorization systems on finite lattices, paying special attention to the role that reflective and coreflective factorization systems play in partitioning the poset of factorization systems on a fixed lattice. We ultimately uncover an intricate web of relations with such diverse combinatorial structures as submonoids, monads, Moore systems, transfer systems (from stable equivariant homotopy theory), and poly-Bernoulli numbers.

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arXiv2024

A Guide to Equivariant Parametrized Cohomology

This article investigates equivariant parametrized cellular cohomology, a cohomology theory introduced by Costenoble-Waner for spaces with an action by a compact Lie group $G$. The theory extends the $RO(G)$-graded cohomology of a $G$-space $B$ to a cohomology graded by $RO(\Pi B)$, the representations of the equivariant fundamental groupoid of $B$. This paper is meant to serve as a guide to this theory and contains some new computations. We explain the key ingredients for defining parametrized cellular cohomology when $G$ is a finite group, with particular attention to the case of the cyclic group $G=C_2$. We compute some examples and observe that $RO(\Pi B)$ is not always free. When $G$ is the trivial group, we explain how to identify equivariant parametrized cellular cohomology with cellular cohomology in local coefficients. Finally, we illustrate the theory with some new computations of parametrized cellular cohomology for several spaces with $G = C_2$ and $G=C_4$.

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arXiv2024

Computational tools for Real topological Hochschild homology

In this paper, we construct a Real equivariant version of the B\"okstedt spectral sequence which takes inputs in the theory of Real Hochschild homology developed by Angelini-Knoll, Gerhardt, and Hill and converges to the equivariant homology of Real topological Hochschild homology, $\text{THR}$. We also show that when $A$ is a commutative $C_2$-ring spectrum, $\text{THR}(A)$ has the structure of an $A$-Hopf algebroid in the $C_2$-equivariant stable homotopy category.

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