Symmetries in toric topology

Fu, Xin (2019) Symmetries in toric topology. University of Southampton, Doctoral Thesis, 124pp.

Record type: Thesis (Doctoral)

Abstract

A polyhedral product (X,A)^K is determined by a finite simplicial complex K and m pairs of topological spaces. The functorial property of polyhedral products implies two types of symmetries of polyhedral products induced by symmetries of simplicial complexes and by group actions on the topological pairs. In this thesis, we consider these two types of symmetries.

In the case of G-polyhedral products induced by a simplicial G-complex K, we show that the homotopy decomposition Sigma (X,A)^K due to Bahri-Bendersky-Cohen-Gitler [3] is homotopy G-equivariant after suspension. It implies a homological decomposition of Hi((X,A)K) in terms of G-representations, which we rely on to study the representation stability of polyhedral product in the sense of Church-Farb [15].

The torus actions on moment-angle complexes Z_K is a special case of actions on polyhedral products induced by actions on the topological pairs. In this thesis, we compute the cohomology of the quotient Z_K/S¹ under free circle actions and introduce a chain complex (C_*(L),δ) whose homology isomorphic to H*(Z_K/S¹;R). For certain cases K, we determine the homotopy types of Z_K/S¹.