# Towards a combinatorial characterization of K(π, 1) arrangements

## Hot Topics: Artin Groups and Arrangements - Topology, Geometry, and Combinatorics March 11, 2024 - March 15, 2024

**Speaker(s):**Masahiko Yoshinaga

**Location:**SLMath: Eisenbud Auditorium, Online/Virtual

**Tags/Keywords**

Artin groups

braid groups

Coxeter groups

reflection groups

Garside structures

classifying spaces

hyperplane arrangements

subspace arrangements

configuration spaces

geometric group theory

Matroids

cohomology

**Primary Mathematics Subject Classification**

**Secondary Mathematics Subject Classification**No Secondary AMS MSC

#### Towards a combinatorial characterization of K(π, 1) arrangements

**A system of half-spaces is an assignment of a half-space to each hyperplane of a real hyperplane arrangement. Such a system is said to be consistent if the intersection of the half spaces is nonempty (a chamber). Given a locally consistent system of half-spaces, using the Salvetti complex, one can construct an embedded sphere in the complexified complement. As a main result, we will show that if the system is (globally) inconsistent, then the resulting sphere gives a non-trivial element in the homotopy group, in particular, it is on-$K(\pi, 1)$ arrangement. This construction recovers many known non-$K(\pi, 1)$ arrangements. We also formulate a conjectural characterizati**on of $K(\pi, 1)$ arrangements. **K(\pi, 1)$ arrangements. We also formulate a conjectural characterization of $K(\pi, 1)$ arrangements**.

#### Towards a combinatorial characterization of K(π, 1) arrangements

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