PhD Defense Mikala Ørsnes Jansen

Title: THE STRATIFIED HOMOTOPY TYPE OF THE REDUCTIVE BOREL–SERRE COMPACTIFICATION AND APPLICATIONS TO ALGEBRAIC K-THEORY

In this thesis, we study the stratified homotopy type of the reductive Borel-Serre compactification, and we investigate how the stratified homotopy type can be used to model unstable algebraic K-theory. The thesis consists of an introduction and two papers, the second of which is joint with Dustin Clausen.

In the first paper, Paper I, we determine the exit path ∞-category of the reductive Borel-Serre compactification of a locally symmetric space associated with a neat arithmetic group. We show that the exit path ∞-category is equivalent to the nerve of a 1-category defined in purely algebraic terms. We derive some immediate corollaries of our result: the homotopy type and, in particular, the fundamental group of the reductive Borel-Serre compactification is determined, and we obtain an identification of the constructible derived category as a derived functor category. To make this identification, we develop several calculational tools applicable to a larger class of stratified spaces.

In the second paper, Paper II, we generalise the exit path 1-category of the reductive Borel-Serre compactification to general linear groups over arbitrary rings, and we show that for a certain class of rings, these categories provide models for unstable algebraic K-theory. For finite fields, the model is in a certain sense better than that given by the plus-construction. We reprove the main result of Paper I using entirely different techniques, and crucially use this proof strategy when dealing with the generalisations. We also define a further generalisation of the exit path 1-category, associating a strict monoidal category to any exact category. We show that these monoidal categories define models for the algebraic K-theory space of exact categories.

 Link to thesis.

 Link to the Zoom invitation: https://ucph-ku.zoom.us/j/63949315412

Adviser: Søren Galatius, University of Copenhagen, Denmark

Assessment committee:

Lars Hesselholt (chairman), University of Copenhagen, Denmark

Dan Petersen,Stockholm University, Sweden

Leslie Saper, Duke University, USA