The Exit Path ∞-Category of the Reductive Borel-Serre Compactification
Event details
| Date | 28.04.2020 |
| Hour | 10:15 › 11:15 |
| Speaker | Mikala Ørsnes Jansen, Københavns Universitet |
| Location | |
| Category | Conferences - Seminars |
For neat arithmetic groups Γ ≤ SLn(ℤ), the locally symmetric space X associated with Γ provides a nice model for the classifying space BΓ : it is a smooth manifold and thus allows for the study of the discrete group to move into the geometric realm. Unfortunately, X is very rarely compact. To remedy this, Borel and Serre in 1973 constructed a compact manifold with corners into which X embeds as the interior. It is now known as the Borel-Serre compactification of X, and it enabled Borel to calculate the ranks of the algebraic K-groups Ki(ℤ). For some purposes, however, the Borel-Serre compactification is ``too big''. Motivated by an interest in L2-cohomology, Zucker introduced another compactification of X in 1982, later coined the reductive Borel-Serre compactification. It is defined as a quotient of the Borel-Serre compactification and is no longer a manifold with corners. It does, however, come equipped with a natural stratification. We set out to understand this stratified space by determining its exit path ∞-category. This is an analogue for stratified spaces of the fundamental ∞-groupoid for topological spaces: it provides information not only about the individual strata but also about how these strata are ``glued'' together. We show that the reductive Borel-Serre compactification is in some sense a K(π,1) of stratified spaces by showing that its exit path ∞-category is equivalent to the nerve of a 1-category. Moreover, some interesting questions arise when looking back at algebraic K-theory.
This is joint work with Dustin Clausen.
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