My research is in topology and geometry—specifically, surgery theory and geometric group theory. A few of my favorite things include: aspherical manifolds, rational homotopy types of manifolds, group actions on manifolds, quantified versions of classical invariants.

I have a list of my published papers.

Finiteness properties

A combination of Bestvina–Brady Morse theory and an acyclic reflection group trick produces a torsion-free finitely presented $\mathbb{Q}$-Poincar'e duality group which is not the fundamental group of an aspherical closed ANR $\mathbb{Q}$-homology manifold.

The acyclic construction suggests asking which $\mathbb{Q}$-Poincar'e duality groups act freely on $\mathbb{Q}$-acyclic spaces (i.e., which groups are $\mbox{FH}(\mathbb{Q})$). For example, the orbifold fundamental group $\Gamma$ of a good orbifold satisfies $\mathbb{Q}$-Poincar'e duality, and we show $\Gamma$ is $\mbox{FH}(\mathbb{Q})$ if the Euler characteristics of certain fixed sets vanish.

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Ph.D. Thesis

We say that a group $G$ is Q-PD if it satisfies Poincare duality with rational coefficients (i.e., if its classifying space $BG$ does). Examples include the fundamental groups of aspherical manifolds.

But there are other geometric examples: if a group $G$ acts freely on a rationally-acyclic, rational homology manifold, then G is Q-PD. Does every Q-PD arise in this way—does every Q-PD group act on such an object?

The answer is no: lattices with torsion in semisimple Lie groups are counterexamples.

Bounded homotopy theory

Given a bounding class $\B$, we construct a bounded refinement ${\BK}(-)$ of Quillen’s $K$-theory functor from rings to spaces. As defined, ${\BK}(-)$ is a functor from weighted rings to spaces, and is equipped with a comparison map $BK \to K$ induced by “forgetting control”. In contrast to the situation with $\B$-bounded cohomology, there is a functorial splitting ${\BK}(-) \simeq K(-) \times {\BK}^{rel}(-)$ where ${\BK}^{rel}(-)$ is the homotopy fiber of the comparison map.

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