Redbud Topology

Diffeomorphism groups in low dimensions

We give a brief overview of smooth manifolds and their diffeomorphism groups in dimensions up to 3. This will include a crash course in the classification of 3-manifolds, including the prime decomposition, JSJ decomposition, and ultimately, geometrization.

The prime decomposition fiber sequence for a reducible 3-manifold

Geometrization is a powerful principle which has shaped our understanding of irreducible 3-manifolds and, more recently, the structure of their diffeomorphism groups. However, diffeomorphism groups of reducible 3-manifolds remain somewhat elusive. Inspired by an approach of Hatcher, we construct a “splitting map” from the classifying space of the diffeomorphism group of a reducible 3-manifold to that of its irreducible prime factors. This results in a “prime decomposition fiber sequence” whose fiber we describe explicitly as a homotopy colimit over a finite category of graphs. We also develop a framework for computation, which we apply to calculate the rational cohomology ring of the classifying space in specific examples. This is joint work with R. Boyd and J. Steinebrunner.

Rigidity of moduli spaces and mapping class groups.

The mapping class group of a surface S is the homeomorphism group of S up to homotopy. In analogy with Margulis superrigidity, homomorphisms between mapping class groups have been classified under certain conditions, and are expected in general to be described in terms of the topology of the surfaces. In the talk we will outline the history and recent progress of this problem, and discuss its connection to the holomorphic rigidity of moduli spaces.

Between cohomology of moduli spaces and polynoimal functors on free groups

The cohomology of moduli spaces of curves with marked points is one of the central and elusive goals of algebraic geometry. One approach to computation passes through a natural quotient with a combinatorial description, related to tropical geometry - the top weight quotient. We will discuss another description of this quotient, relating it to "representations" of the category of free groups, and leading to calculations of a large part of the cohomology with a geometric flavor.

Homological stability and simplicial complexes

After briefly reviewing the definition of group homology, I will survey homological stability results for general linear groups. I will describe how Borel’s calculations of the stable homology of GL_n(Z) was used by Farrell—Hsian to understand the topology of diffeomorphism groups. I will introduce several simplicial complexes used in homological stability arguments and describe techniques for proving these simplicial complexes are highly connected. While this talk is primarily expository, I will also describe joint work with Bernard and Sroka on improved stable ranges for homological stability.

Homological stability and beyond

I will survey what is know about the homology of GL_n(Z) and its subgroups. This includes many structural results and conjectures about the homology in low and high homological degrees. For GL_n(Z), these patterns manifest as homological stability in low degrees and homological vanishing in high degrees. For congruence subgroups, the analogous patterns are forms of representation stability. This includes joint work with Ash, Brück, Kupers, Nagpal, Patzt, Putman, Sroka, and Wilson.

Coarse homological invariants of metric spaces

In this talk, we will introduce several coarse topological invariants of metric spaces, inspired by and analogous to, classical concepts in the study of group cohomology. Using the notion of an R-module over a metric space, we can interpret group cohomological notions, such as finiteness properties and cohomological dimension, for arbitrary metric spaces. Extending a result of Sauer, it is shown that coarse cohomological dimension is monotone under coarse embeddings, and hence is invariant under coarse equivalence. Time permitting, we will discuss a higher dimensional analog of classical the Hopf-Freudenthal theorem that a group has either 0, 1, 2 or infinitely many ends.

The Second Rational Homology of the Torelli Group

The Torelli group of a surface is the subgroup of the mapping class group that acts trivially on the first homology of the surface. In joint work with Andrew Putman, we compute the second rational homology of the Torelli group for all surfaces of sufficiently high genus. In particular, this vector space is finite dimensional and algebraic for all surfaces of genus at least 6.

The top cohomology of principal congruence subgroups of special linear and symplectic groups

I will discuss the rational cohomology of special linear and symplectic groups, and their principal congruence subgroups, for number rings R. Borel–Serre showed that these groups satisfy a (co)homological duality that lets us study their cohomology groups via certain representations called the `Steinberg modules’, which have a combinatorial description in terms of Tits buildings. I will describe forthcoming work that uses this approach to compute the top cohomology of certain principal congruence subgroups of prime level, when R is a Euclidean domain. The computations rely on the structure of the units in the field obtained by quotienting R by the prime, modulo the units coming from R itself.

Translation surfaces, abelian differentials, and their moduli

This will be a very brief introduction to translation surfaces, abelian differentials, and their moduli spaces (“strata”). We will explain the fundamental correspondence between the geometric object known as a translation surface, and the corresponding algebro-geometric object known as an abelian differential, and make some basic remarks about the structure of the space of all such objects.

Fundamental groups of some strata of abelian differentials

Moduli spaces (“strata”) of abelian differentials (aka translation surfaces) are a meeting ground for algebraic geometry, dynamics, flat geometry and more. Strata are closely related to the moduli space of Riemann surfaces, but unlike in the latter case, where the topology is well-understood (it is an orbifold K(G,1) space for the mapping class group), the topology of a given stratum has historically been mysterious. I will report on recent work that determines the (orbifold) fundamental group of any stratum of differentials with enough simple zeroes, identifying it with a certain “framed mapping class group”.

Uniform representation stability for ordered Hurwitz spaces

"Ordered Hurwitz spaces" are moduli spaces of branched covers of the Riemann sphere with a choice of ordering on the branch points. Motivated by applications to number theory, Jordan Ellenberg conjectured that the cohomology of these spaces satisfies a condition called "representation stability" with respect to the action of the permutation group of the branch points. In this talk I will explain this conjecture, and a resolution due to work joint with Z. Himes and J. Miller.