Studying the topology of a fibration sequence frequently involves the monodromy action, which is the action of the fundamental group \(\pi\) of the base space \(B\) on the fibre \(F\). When using a spectral sequence one may need to consider the homology of the base with coefficients in the homology of the fibre regarded as an \(R\pi\)-module, where \(R\pi\) is the group ring. Here, the monodromy representation for a fibration \(p : E\rightarrow B\) with fibre \(F\) means the representation \(\rho : \pi_1(B)\rightarrow \text{Out}(H_1(F))\). The goal of the paper under review is to study polyhedral products in connection with monodromy representations for certain fibrations that arise naturally in the field of toric topology.
In this paper, the author presents a machinery based on polyhedral products that produces faithful representations of graph products of finite groups and direct products of finite groups into automorphisms of free groups \(\text{Aut}(F_n)\) and outer automorphisms of free groups \(\text{Out}(F_n)\), respectively, as well as faithful representations of products of finite groups into the linear groups \(\text{SL}(n, \mathbb{Z})\) and \(\text{GL}(n, \mathbb{Z})\). These faithful representations are realized as monodromy representations.