Homology operations revisited. 47 pages. Submitted for publication. arXiv preprint The mod p homology of E-infinity spaces is a classical topic in algebraic topology traditionally approached in terms of Dyer–Lashof operations. This paper offers a new perspective on the subject by providing a detailed investigation of an alternative family of homology operations equivalent to, but distinct from, the Dyer–Lashof operations. Among other things, we relate these operations to the Dyer–Lashof operations, describe the algebra generated by them, and use them to describe the homology of free E-infinity spaces. We also investigate the relationship between the operations arising from the additive and multiplicative E-infinity structures on an E-infinity ring space. The operations have especially good properties in this context, allowing for a simple and conceptual formulation of “mixed Adem relations” describing how the operations arising from the two different E-infinity structures interact.
String topology of finite groups of Lie type. Joint with Jesper Grodal. 58 pages. arXiv preprint This paper establishes an unexpected connection between finite groups of Lie type and string topology of classifying spaces of compact Lie groups: the cohomology of a finite group of Lie type is a module over the cohomology of the free loop space of the classifying space of the corresponding compact Lie group when the latter cohomology groups are equipped with a string topological multiplication. This module structure gives, among other things, a new and structured way to approach the Tezuka conjecture asserting that under certain conditions, the two cohomologies are isomorphic.
Modular characteristic classes for representations over finite fields. Joint with David Sprehn. Advances in Mathematics, Volume 323, January 2018, Pages 1–37. DOI: 10.1016/j.aim.2017.10.029 article, arXiv preprint The cohomology of the degree-n general linear group over a finite field of characteristic p, with coefficients also in characteristic p, remains poorly understood. For example, the lowest degree previously known to contain nontrivial elements is exponential in n. In this paper, we introduce a new system of characteristic classes for representations over finite fields, and use it to construct a wealth of explicit nontrivial elements in these cohomology groups. In particular we obtain nontrivial elements in degrees linear in n. We also construct nontrivial elements in the mod p homology and cohomology of the automorphism groups of free groups, and the general linear groups over the integers. These elements reside in the unstable range where the homology and cohomology remain poorly understood. The paper was inspired by my previous computations of higher string topology operations.
Higher operations in string topology of classifying spaces. Mathematische Annalen, Volume 368, Issue 1–2, June 2017, Pages 1–63. DOI: 10.1007/s00208-016-1406-1 article, arXiv preprint Examples of non-trivial higher string topology operations have been rare in the literature. This paper ameliorates the situation by providing explicit computations of a wealth of such operations. It also begins the work of applying the string topological methods enabled by my work with Hepworth to the study of automorphism groups of free groups: As an application of the computations, one obtains a wealth of interesting homology classes in the twisted homology groups of automorphism groups of free groups, the ordinary homology groups of holomorphs of free groups, and the ordinary homology groups of affine groups over the integers and the field of two elements. The elements constructed live in the unstable range where the homology of these groups remains poorly understood.
On string topology of classifying spaces. Joint with Richard Hepworth. Advances in Mathematics, Volume 281, August 2015, Pages 394–507. DOI: 10.1016/j.aim.2015.03.022 article, arXiv preprint The main result of the paper is the extension of string topology of classifying spaces into a new kind of field theory which has operations parametrized by the homology groups of automorphism groups of free groups with boundaries in addition to operations parametrized by the homology groups of mapping class groups of surfaces. The paper was the subject of a series of three invited lectures at Loop spaces in geometry and topology, a large international conference held at Centre de Mathématiques Henri Lebesgue, Nantes, France, September 1–5, 2014.
The Atiyah–Segal completion theorem in twisted K-theory. Algebraic & Geometric Topology 12 (2012) 1925–1940. DOI: 10.2140/agt.2012.12.1925 article, arXiv preprint A fundamental result in equivariant K-theory, the classical Atiyah–Segal completion theorem relates the G-equivariant K-theory of a finite G-CW complex X to the non-equivariant K-theory of the Borel construction of X. Here G is a compact Lie group. In this paper, which formed an important part of my PhD research, I prove that the Atiyah–Segal completion theorem also holds in twisted K-theory, a form of K-theory which in recent years has been the focus of intense study because of its connections to string theory.