# Simplicial complexes workshop

There will be one talk each day from 11:00 - 12:00, and we'll go for lunch together afterwards. If you'd like to join meet us in Aud 1 in August Krogh Building at that time (and don't forget to bring a cup of coffee!)

Monday, August 17th 2020, 11:00 - 12:00
On the coset poset of 2-nilpotent subgroups in a 3-nilpotent group (Simon Gritschacher)

To a group G and a family of subgroups F of G one can associate a simplicial complex C(F,G) constructed from the poset of all cosets gH where g \in G and H runs through F. It is a natural question to ask how the topological properties of C(F,G) are related to the algebraic properties of F and G, in particular with regards to higher connectivity of C(F,G). I will look at the particular case where G is a nilpotent group of class at most 3 and F the family of subgroups of class at most 2. For certain G, 1- and 2-connectivity of C(F,G) detect whether G is a 2-Engel or 2-nilpotent group, respectively. This addresses a question of Okay and is based on discussions with Bernardo Villarreal.

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Tuesday, August 18th 2020, 11:00 - 12:00
Discrete Morse Theory and E^k buildings (Peter Patzt)

In this talk, I will recall discrete Morse theory and then apply it to sketch why the E^2 building T^2_n is (2n-3)-spherical for Euclidean rings. This is early-state work-in-progress with Alexander Kupers, Jeremy Miller, and Jennifer Wilson with potential applications to algebraic K-theory of the integers.
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Wednesday, August 19th 2020, 11:00 - 12:00
Coset complexes (Benjamin Brück)

Coset complexes provide one with a possibility to construct for a given group G simplicial complexes that are equipped with an action of G and have a prescribed set of simplex stabilisers. These complexes have nice combinatorial and topological properties: They are completely balanced, facet transitive and whether they are connected or simply connected can easily be expressed in terms of the simplex stabilisers of the G-action. Examples of this construction include Coxeter complexes (G = a Coxeter group), Tits buildings (G = GL_n or more generally a group with BN pair) and the free factor complex (G = Aut(F_n) ).
The talk will be a short introduction to this family complexes. I will present some of their properties, give examples and in particular talk about the case of coset complexes that are Cohen-Macaulay.

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Thursday, August 20th, 11:00 - 12:00
Centrally symmetric complexes and the minimal triangulation of RP^d (Hailun Zheng)

Abstract: Let n(d) denote the minimum number of vertices required to triangulate the d-dimensional real projective space. Arnoux and Marin showed that n(d) \geq {d+2 choose 2}+1 for d \geq 3, while Kühnel proved that n(d) \leq 2^{d+1}-1. Which bound is more accurate? In this talk, I will discuss several lower dimensional constructions and a recent result of Venturello and I that improves Kühnel’s upper bound. On a related note, I will showcase a few properties of centrally symmetric complexes that more or less explain why finding a "better" upper and lower bound of n(d) is not as easy as it seems to be.

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Friday, August 21st, 11:00 - 12:00
Algebraic analogues of coset posets and the low dimensional homology of Iwahori-Hecke algebras (Robin Sroka)

Iwahori-Hecke algebras are a family of algebras associated to the choice of a Coxeter group and a commutative ring, an example being the group algebra of the Coxeter group. Every Coxeter group acts on a contractible space called the Davis complex, which is the geometric realization of a coset poset. In this talk, we introduce an algebraic analogue of the Davis complex for Iwahori-Hecke algebras and explain how it can be used to compute the low dimensional homology of Iwahori-Hecke algebras, interpreted as appropriate Tor-groups. The group algebra case recovers known computations of the low dimensional homology of Coxeter groups. This is a work in progress inspired by conversations with Hepworth, a paper of Boyd and their recent joint work.