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.
You may also be interested in seeing a list of my publications or reading my research statement.
Projective planes Joint work with Zhixu Su
The Hirzebruch L-polynomial is one place where number theory very strongly interacts with high-dimensional topology . Recall that the Hirzebruch signature theorem relates the signature of a smooth closed manifold M4k to ∑ILIpI(M). Unfortunately, the naïve method to compute coefficients LI of the Hirzebruch L-polynomial is much too slow for applications; Zhixu Su and I have discovered a recursive method which is fast enough to compute many coefficients. Solutions to some Diophantine equations related to these L-polynomials give rise to manifolds having a truncated polynomial algebra as their rational cohomology ring; such manifolds may exist even when the corresponding truncated polynomial algebra over ℤ is not the cohomology ring of any space. For instance, there is a manifold having the rational cohomology that 𝕆P4 would be expected to have, if 𝕆P4 existed.
Aspherical manifolds that cannot be triangulated Joint work with Michael W. Davis, Jean-François Lafont
Kirby and Siebenmann showed that there are manifolds that do not admit PL structures, and yet the possibility remained that all manifolds could be triangulated, meaning that for every manifold M, there is a simplicial complex K so that the geometric realization of K is homeomorphic to M, but of course the simplicial complex K is not a PL triangulation, meaning the links are not spheres. Freedman showed that there are 4-manifolds that cannot be triangulated. Davis and Januszkiewicz applied a hyperbolization procedure to Freedman’s 4-manifolds to get closed aspherical 4-manifolds that cannot be triangulated. What about higher dimensions?
The no-three-in-line problem on a torus Joint work with Andrew Groot, Deven Pandya, Bart Snapp
For a group G, let T(G) denote the cardinality of the largest subset S ⊂ G so that no three elements of S are in the same coset of a cyclic subgroup. Undergraduates Andrew Groot and Deven Pandya, advised by myself and my colleague Bart Snapp, considered the case G = ℤ/mℤ × ℤ/nℤ, and showed that T(ℤp × ℤp2) = 2p and T(ℤp × ℤpq) = p + 1.
A combination of Bestvina–Brady Morse theory and an acyclic reflection group trick produces a torsion-free finitely presented ℚ-Poincaré duality group which is not the fundamental group of an aspherical closed ANR ℚ-homology manifold.
Computation of the total L-class would also solve some manifold recognition problems, such as recognizing whether a particular combinatorial 15-vertex triangulations of an 8-manifold is the quaternionic projective plane ℍP8. One of these examples X8 is especially symmetric, and likely PL homeomorphic to ℍP8.
Bounded homotopy theory Joint work with Crichton Ogle
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 → K induced by “forgetting control.” In contrast to the situation with ℬ-bounded cohomology, there is a functorial splitting ℬK( − ) ≃ K( − ) × ℬKrel( − ) where ℬKrel( − ) is the homotopy fiber of the comparison map.
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.
Cusp size bounds from singular surfaces in hyperbolic 3-manifolds Joint work with Colin Adams, Adam Colestock, William Gillam, Eric Katerman
Singular maps of surfaces into a hyperbolic 3-manifold are utilized to find upper bounds on meridian length, ℓ-curve length and maximal cusp volume for the manifold. This allows a proof of the fact that there exist hyperbolic knots with arbitrarily small cusp density and that every closed orientable 3-manifold contains a knot whose complement is hyperbolic with maximal cusp volume less than or equal to 9.
Cleanliness of geodesics in hyperbolic 3-manifolds Joint work with Colin Adams, Adam Colestock, William Gillam, Eric Katerman
We derive conditions guaranteeing the existence of geodesics avoiding the cusps and use these geodesics to show that in “almost all” finite volume hyperbolic 3-manifolds, infinitely many horoballs in the universal cover corresponding to a cusp are visible in a fundamental domain of the cusp when viewed from infinity.