One-relator groups and algebras related to polyhedral products

Abstract

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$ , we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal {R}_K$ , to be a one-relator group; and for the Pontryagin algebra $H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For $RC_K'$ , it is given by a condition on the homology group $H_2(\mathcal {R}_K)$ , whereas for $H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal {Z}_K$ .

Pontryagin Algebras and the LS-Category of Moment–Angle Complexes in the Flag Case

“…, and hence H * (ΩZ K ; k) is a one-relator algebra. It was considered in [Ver16,GIPS21]. From the point of view of Theorem 5.5, the relation corresponds to the 1-cycle…”

Adams-Hilton models and higher Whitehead brackets for polyhedral products

“…the formula (4.9) holds. By Theorem 4.3 (10), we may choose the maps ∂ and θ on AH(CP k 1 × • • • × CP kn ) inductively in such a way that the ring homomorphism…”

“…Adams-Hilton models for S 5 × S 2 and T (S 2 , S 2 , S 2 ) × S 2 are described in Theorem 4.10 and Corollary 4.11. It is easy to see that the maps of algebras defined in Theorem 4.3 (10) AH(S 5 × S 2 )…”

“…The homology of the Adams-Hilton model is the Pontryagin algebra H * (ΩX). Adams-Hilton models have already arisen in the study of polyhedral products [12,10]. There is an indeterminacy in the construction of AH(X), so there is a family of Adams-Hilton models for a given X.…”

“…3(10), there is an Adams-Hilton model (AH(CP n × CP k ), ∂) extending the model AH(sk 2N −2 (CP n × CP k )) and satisfying ν∂ = ∂ν.Consider the generator χ σ , σ = {1, . .…”