IMUNAM Algebraic Topology Seminar

Title: (Talk in spanish) Propiedades de finitud de espacios móduli de variedades suaves.

Abstract: El espacio móduli BDiff(M) de una variedad suave M es el espacio que clasifica haces fibrados con fibra M. Entender su tipo de homotopía es entonces la tarea que tiene todo topólogo interesado en clasificar haces fibrados. Para bien o para mal, estos espacios tienden a ser bastante complicados, por lo cual nos toca conformarnos con tratar de calcular algunos de sus invariantes topológicos. Esto tampoco es tarea fácil, incluso en los casos más simples (eg. M una esfera) son pocas las cuentas que hay disponibles. Sin embargo, hay algo un poco más "cualitativo" que sí se puede afirmar en mucha generalidad: si M es una variedad suave compacta conexa de dimensión par mayor que 5 y con grupo fundamental finito, entonces todos los grupos de homotopía y homología del espacio móduli BDiff(M) son finitamente generados. En esta charla voy a discutir algunas de las ideas que M. Krannich, A. Kupers y yo usamos en la demostración de ésta afirmación.

Title: An (∞, 2)-categorical pasting theorem

Abstract: Power's 2-categorical pasting theorem, asserting that any pasting diagram in a 2-category has a unique composite, is at the basis of the 2-categorical graphical calculus, which is used extensively to develop the theory of 2-categories. In this talk we discuss an (∞, 2)-categorical analog of the pasting theorem, asserting that the space of composites of any pasting diagram in an (∞, 2)-category is contractible. This result, which is joint with Hackney—Ozornova—Riehl, rediscovers independent work by Columbus.

Title: Dynamical Induction and Cohomology of Crystallographic Groups

Abstract: Adem, Lueck, and their collaborators have shown that in various situations, the Lyndon-Hochschild-Serre spectral sequence associate to a crystallographic group collapses at the second page. Crystallographic groups are built from integral representations finite groups. We describe a generalized notion of induction, which we call dynamical induction, originating in the theory of C*-algebras, and we show that these collapse results are preserved under induction in an appropriate sense. In particular, we show that the spectral sequence associated to a semidirect product of the form Z^n \rtimes Q collapses whenever the action of Q on Z^n is induced up from an action of a group of square-free order. This is joint work with my recent Ph.D. student, Chris Neuffer.

Title: Stratifying integral representations of finite groups.

Abstract: Classifying all integral representations of finite groups is essentially impossible. In this talk, we will introduce an integral version of the stable module category for a finite group G and then explain how to use it to give a 'generic' classification of integral G-representations. Our results globalize the modular case established by Benson, Iyengar, and Krause and relies on the notion of stratification in tensor triangular geometry developed in joint work with Heard and Sanders. Time permitting, I will discuss some further applications.

Title: Towards Directed Collapsibility

Abstract: While collapsibility of CW complexes dates back to the 1930s, collapsibility of directed complexes has not been well studied to date. We define a notion of directed collapsibility in the setting of a directed Euclidean cubical complex that builds on the classical definition of collapsibility. We call this type of collapse a link-preserving directed collapse. In the undirected setting, pairs of cells are removed from a space if a deformation retract remains. In the directed setting, topological properties - in particular, properties of spaces of directed paths - would not always be preserved. The direction of the space can be taken into account by requiring that the past links of vertices remain homotopy equivalent after collapsing. We show that there are computationally simple conditions which preserve the topology of past links. Furthermore, we give conditions for when link-preserving directed collapses preserve the contractability and connectedness of spaces of directed paths. Our results have applications to speeding up the verification process of concurrent programming and to understanding partial executions in concurrent programs. This is joint work with Robin Belton, Robyn Brooks, Stefania Ebli, Lisbeth Fajstrup, Brittany Terese Fasy, and Nicole Sanderson.

Absract: This is joint work with Jeremy Miller and Jennifer Wilson. Let M be an orientable surface of genus g with one boundary curve, and let F_n(M) denote the configuration space of n ordered points in M. The action of Homeo(M,dM) on F_n(M) descends to an action of the mapping class group Gamma(M,dM) on the homology H_*(F_n(M)). Our main result is that, for all n,i>=0, the i-th stage J(i) of the Johnson’s filtration of Gamma(M,dM) acts trivially on H_i(F_n(M)). This extends previous work of Moriyama on certain relative configuration spaces. I will recall the necessary definitions and give a sketch of the proof of the main theorem: the main inputs are Moriyama’s work and a cell stratification of F_n(M) à la Fox-Neuwirth-Fuchs. I will also present some examples of non-trivial actions of mapping classes in J(i-1) on elements of H_i(F_n(M)), for small values of i.

Title: High-dimensional cohomology of special linear and symplectic groups

Abstract: Computing the cohomology of arithmetic groups is a fundamental and often difficult problem at the intersection of topology, group theory and number theory. In this talk, I will explain how one can use duality phenoma to compute the rational cohomology of the arithmetic groups SL_n(Z) and Sp_2n(Z) in "high" dimensions, i.e. close to their virtual cohomological dimension. Specifically, I will talk about joint work with Miller-Patzt-Sroka-Wilson in which we show that H^{n(n-1)/2 - 2}(SL_n(Z); Q) = 0 for n>3. This was previously unknown, but confirms a conjecture of Church-Farb-Putman. In ongoing work with Patzt-Sroka, we are also trying to adapt these techniques to the group Sp_2n(Z).

Title: Smoothing fibre bundles and diffeomorphism groups of discs

Abstract: Given a fibre bundle p: E -> B of topological manifolds with d-dimensional fibres, one might ask whether p is fibrewise homeomorphic to a smooth bundle. If d≠4, then there is a strategy to answer this question based on smoothing and obstruction theory. The main limiting factor in this approach is that it requires knowledge of the homotopy groups of the topological group of diffeomorphisms of a closed d-disc. The study of these homotopy groups has quite a history in geometric topology and turns out to be related to algebraic K- and L-theory, stable homotopy theory, and the combinatorics of finite graphs. In this talk I will explain the relation between smoothing fibre bundles and the homotopy groups of diffeomorphisms of discs, survey the state of the art in the computation of the latter, and explain aspects of recent joint work with Oscar Randal-Williams.