On monodromy representations in Denham–Suciu fibrations

Abstract

Communicated by C.A. Weibel MSC: 55U10; 58K10; 13F55; 14F45We study the monodromy representation corresponding to a fibration introduced by G. Denham and A. Suciu [5], which involves polyhedral products given in Definition 2.2. Algebraic and geometric descriptions for these monodromy representations are given. In particular, we study the case of a product of two finite cyclic groups and obtain representations into Out(F n ) and SL n (Z). We give algebraic descriptions of monodromy for the case of a product of any two finite groups. Finally we give a geometric description for monodromy representations of a product of 2 or more finite groups to Out(F n ), as well as some algebraic properties. The geometric description does not rely on choosing a basis for the fundamental group of the fibre in terms of commutators, hence avoids this delicate question.

Polyhedral products, flag complexes and monodromy representations

“…For examples concerning only two finite groups, i.e. n = 2, see [23], where explicit answers are given. We can explicitly describe faithful representations (eg.…”

“…In [23,Theorem 2.2] it was shown that two cyclic subgroups yield a faithful monodromy representation…”

“…The goal of this paper is to study polyhedral products in connection with monodromy representations for certain fibrations that arise naturally in the field of toric topology. The problem of explicitly describing such representations was studied by the author in [23].…”

“…, EG n }, see Definition 2.1. We give explicit descriptions of monodromy representations for simplicial complexes K with more than two vertices, which were described geometrically in [23]. To do this we generalize and use some results of Panov and Veryovkin [20].…”

A Survey on Spaces of Homomorphisms to Lie Groups

“…For example, if G is a finite group of odd order, then the classical Feit-Thompson theorem is equivalent to the map H 1 (E(2, G)) G → H 1 (B(2, G)), where H 1 (E(2, G)) G denotes the module of coinvariants under the monodromy action, failing to be surjective. Properties of the monodromy representation may be interesting, but are currently not well understood [2,28]. This setting suggests that the spaces B(2, G) contain compelling information.…”