MathOverflow

I'm preparing a talk around Kensuke Arakawa's Classifying Space via Homotopy Coherent Nerve and would like concrete applications of classifying spaces of simplicial groups.
Let $G \in \mathrm{Cat}_\Delta$ be a simplicial group, i.e. a simplicially enriched category with a single object $*$ and each $n$-simplex invertible in $G(*,*)$.Arakawa proves that the homotopy coherent nerve $NG$ given by$$ [n] \mapsto \mathrm{Cat...

I am interested in a polygonal domain $\Omega$ in the complex plane. For simplicity, we may assume that its closure $\overline{\Omega}$ is compact (so it has no edges going to infinity, for example). Let the vertices be $z_i$ for $i = 1, \dots, n$. I...

Well known structure theorems for abelian locally compact groups show that every abelian locally compact group can be constructed from the groups $\mathbf R$, $\mathbf Z$, $\mathbf T$ and cyclic groups using the operations of finite products, projective...

Disclaimer(Cross-post): The question is somewhat related to the unanswered ms question 1
Problem:Consider $x_1,\ldots,x_6\in \mathbb{R}^3$ such that $\sum_{j=1}^6\|x_j\|^2=3$. I want to prove the following: $$\sum_{1\leq i<j<k \leq 6}|\det(x_i,x_j,x_k)|\leq \sqrt{10+4\sqrt5},$$with equality occuring when $\pm x_1,\ldots,\pm x_6$...