# Ben Williams

I am an Associate Professor at UBC in Mathematics. My research interests encompass the application of homotopy-theoretic methods to problems in, or inspired by, algebra and algebraic geometry.

My office is MATX 1205 (this is in UBC's Math Annex building, near the main math building).

Here is my CV.

## Teaching

In the second winter term of 2023-2024, I am teaching Math 220 (Mathematical Proof) and Math 221 (Matrix Algebra). Both these courses have Canvas websites, which students of those courses should consult.

#### Math 221

I have frequently taught Math 221, Matrix Algebra. Here is a free online textbook from this course. The textbook is a fork of the excellent book by Dan Margalit and Joseph Rabinoff.

#### Math 426

I have frequently taught Math 426, Introduction to Topology. Here is the course website from the last time I taught this course. It contains my lecture notes, which may be useful to someone.

## Papers

#### Spaces of generators for matrix algebras with involution

Joint with Taeuk Nam and Cindy Tan, in Linear and Multilinear Algebra.

#### On the minimal number of generators of an étale algebra

Joint with Abhishek Kumar Shukla. This appears in the Canadian Journal of Mathematics.

#### Motivic spheres and the image of Suslin's Hurewicz map

Joint with Aravind Asok and Jean Fasel.

#### The simplicial suspension sequence in A1-homotopy

Joint with Aravind Asok and Kirsten Wickelgren

Geometry and Topology 21 (2017)

We study a version of the James model for the loop space of a suspension in unstable A1–homotopy theory. We use this model to establish an analog of G W Whitehead’s classical refinement of the Freudenthal suspension theorem in A1–homotopy theory: our result refines F Morel’s A1–simplicial suspension theorem. We then describe some E1–differentials in the EHP sequence in A1–homotopy theory. These results are analogous to classical results of G W Whitehead. Using these tools, we deduce some new results about unstable A1–homotopy sheaves of motivic spheres, including the counterpart of a classical rational nonvanishing result.

#### Prime decompostion for the index of a Brauer class

Joint with Ben Antieau

Annali della Scuola Normale Superiore di Pisa—Classe di scienze, 17 (2017) 277–285

#### Topology and Purity for torsors

Joint with Ben Antieau

Documente Mathematica 20 (2015)

We study the homotopy theory of the classifying space of the complex projective linear groups to prove that purity fails for $\mathrm{PGL}_p$-torsors on regular noetherian schemes when $p$ is a prime. Extending our previous work when $p=2$, we obtain a negative answer to a question of Colliot-Thélène and Sansuc, for all $\mathrm{PGL}_p$. We also give a new example of the failure of purity for the cohomological filtration on the Witt group, which is the first example of this kind of a variety over an algebraically closed field.

#### The prime divisors of the period and index of a Brauer class

Joint with Ben Antieau

Journal of Pure & Applied Algebra

This paper exists in a concise form, available at the arXiv link above and, we expect, in publication. It also has an extended version, the additions being largely notes for our own benefit on topoi.

#### The period-index problem for twisted topological $K$-theory

Joint with Ben Antieau

Geometry & Topology 18 (2014) 1115–1148

This paper was the first-started paper in the series of joint papers with Ben Antieau on the period–index problem and related topics. The earlier-appearing papers in this series cite it in preprint form.

We introduce and solve a period-index problem for the Brauer group of a topological space. The period-index problem is to relate the order of a class in the Brauer group to the degrees of Azumaya algebras representing it. For any space of dimension $d$, we give upper bounds on the index depending only on $d$ and the order of the class. By the Oka principle, this also solves the period-index problem for the analytic Brauer group of any Stein space that has the homotopy type of a finite CW-complex. Our methods use twisted topological $K$-theory, which was first introduced by Donovan and Karoubi. We also study the cohomology of the projective unitary groups to give cohomological obstructions to a class being represented by an Azumaya algebra of degree $n$. Applying this to the finite skeleta of the Eilenberg-MacLane space $K(Z/\ell,2)$, where $\ell$ is a prime, we construct a sequence of spaces with an order $\ell$ class in $\operatorname{Br}$, but whose indices tend to infinity.

#### Unramified division algebras do not always contain Azumaya maximal orders

Joint with Ben Antieau

Invent. Math. (2013)

We show that, in general, over a regular integral noetherian affine scheme $X$ of dimension at least 6, there exist Brauer classes on $X$ for which the associated division algebras over the generic point have no Azumaya maximal orders over $X$. Despite the algebraic nature of the result, our proof relies on the topology of classifying spaces of algebraic groups.

#### Godeaux–Serre varieties and the étale index

Joint with Ben Antieau

The order in which the names ‘Godeaux’ and ‘Serre’ appear in the title of this paper was reversed shortly before publication. The arXiv version retains the old ordering.

We use Godeaux-Serre varieties of finite groups, projective representation theory, the twisted Atiyah-Segal completion theorem, and our previous work on the topological period-index problem to compute the étale index of Brauer classes $\alpha \in \mathrm{Br}_{\text{et}}(X)$ in some specific examples. In particular, these computations show that the étale index of $\alpha$ differs from the period of $\alpha$ in general. As an application, we compute the index of unramified classes in the function fields of high-dimensional Godeaux-Serre varieties in terms of projective representation theory.

#### On the classification of oriented $3$-plane bundles over a 6-complex

Joint with Ben Antieau

This short note corrects a minor error in L. M. Woodward's 1982 paper "The classification of principal $\mathrm{PU}_n$-bundles over a 4-complex."

#### The topological period–index problem over 6–complexes

Joint with Ben Antieau.

J. Top. 7 (2014), 617-640.

By comparing the Postnikov towers of the classifying spaces of projective unitary groups and the differentials in a twisted Atiyah-Hirzebruch spectral sequence, we deduce a lower bound on the topological index in terms of the period, and solve the topological version of the period-index problem in full for finite CW complexes of dimension at most 6. Conditions are established that, if they were met in the cohomology of a smooth complex 3-fold variety, would disprove the ordinary period-index conjecture. Examples of higher-dimensional varieties meeting these conditions are provided. We use our results to furnish an obstruction to realizing a period-2 Brauer class as the class associated to a sheaf of Clifford algebras, and varieties are constructed for which the total Clifford invariant map is not surjective. No such examples were previously known.

#### The $\mathbb{G}_m$-equivariant Motivic Cohomology of Stiefel Varieties.

We derive a version of the Rothenberg-Steenrod, fiber-to-base, spectral sequence for cohomology theories represented in model categories of simplicial presheaves. We then apply this spectral sequence to calculate the equivariant motivic cohomology of the general linear group with a general $\mathbb{G}_m$–action, this coincides with the equivariant higher Chow groups. Some of the equivariant motivic cohomology of a Stiefel variety with a general $\mathbb{G}_m$–action is deduced as a corollary.

#### The Motivic Cohomology of Stiefel Varieties

The main result of this paper is a computation of the motivic cohomology of varieties of $n \times m$-matrices of rank $m$, including both the ring structure and the action of the reduced power operations. The argument proceeds by a comparison of the general linear group-scheme with a Tate suspension of a space which is $\mathbb{A}^1$-equivalent to projective $n-1$-space with a disjoint basepoint.

## Preprints, Notes & Other Material

#### Connectivity of Manifold Complements

This note is dedicated to proofs of two well-known facts whose proofs I did not find in the literature. First: the homotopy groups of smooth manifolds may be calculated using smooth maps and smooth homotopies between them, and second: that if $M$ is a smooth manifold and $Z$ is a closed submanifold of codimension $d$, then the inclusion map $M \backslash Z \hookrightarrow M$ is $d-1$-connected. If you know of proofs of these facts in textbooks or published papers, please let me know.

#### A classification of symmetries of knots

Joint with Keegan Boyle and Nicholas Rouse. Submitted for publication.

#### Spaces of Generators for the 2×2 Matrix Algebra

Joint with Sebastian Gant. Submitted for publication.

#### Unstable Motivic Homotopy Theory

Joint with Kirsten Wickelgren. This is an expository article on $\mathbb{A}^1$– or motivic–homotopy, and contains no original research. It appears in the Handbook of Homotopy Theory edited by Haynes Miller.

#### The Equivariant Motivic Cohomology of Varieties of Long Exact Sequences.

My PhD Thesis.

This thesis is in four chapters. The results of the first are more elegantly stated in The Motivic Cohomology of Stiefel Varieties above, and the results of the second & third were expanded upon and presented in The $\mathbb{G}_m$-equivariant Motivic Cohomology of Stiefel Varieties above. The material of the fourth chapter has not yet been written into a paper.