Abstract:
Using results on the topology of polyhedral products, we link distinct concepts of homotopy theory and geometric group theory. On the homotopical side, we describe the Pontryagin algebra (loop homology) of the moment-angle complex $\mathcal Z_K$. On the group-theoretical side, we describe the structure of the commutator subgroup $RC_K'$ of a right-angled Coxeter group $RC_K$, viewed as the fundamental group of the real moment-angle complex $\mathcal R_K$. For a flag simplicial complex $K$, we present a minimal generator set for the Pontryagin algebra $H_*(\Omega\mathcal Z_K)$ and for the commutator subgroup $RC_K'$, and specify a necessary and sufficient combinatorial condition for $H_*(\Omega\mathcal Z_K)$ and $RC_K'$ to be a free or one-relator algebra (group). We also give homological characterisations of these properties. For $RC_K'$, this is given by a condition on the homology group $H_2(\mathcal R_K)$, whereas for $H_*(\Omega \mathcal Z_K)$ this can be stated using the bigrading of the homology of $\mathcal Z_K$.
Parts of this talk are joint works with Jelena Grbic, Marina Ilyasova, George Simmons, Stephen Theriault, Yakov Veryovkin and Jie Wu.