Equivariance, Module Structure, Branched Covers, Strickland Maps and Cohomology Related to the Polyhedral Product Functor

Abstract

This thesis will be directed towards the analysis of a construction known as the polyhedral product functor. The polyhedral product functor is a subspace of a product space obtained by gluing certain topological spaces in a combinatorial way dictated by a simplicial complex K[Buchstaber and Panov, 2000]. The thesis will be divided into 6 parts as follows.

In the first part, we study the equivariance of the stable decomposition formula for the polyhedral product functor Z_K( X,A) given by A. Bahri, M. Bendersky, F. R. Cohen and S. Gitler [Bahri et al., 2010],}, under the action of Aut( K), which is the automorphism group of the simiplicial complex K. In the second part, we analyze the module structure of H*(Σ(Z_Kn(D ¹,S⁰))), for K _n the boundary of the n-gon and Σ(Z_Kn( D¹,S⁰)) is the reduced suspension of Z_Kn(D¹, S⁰) as defined in 1.0.5, under the action of the cyclic group C_n. induced by the action of the cyclic group C_n on the n-gon K_n. In part 3 we introduce branched covers of the polyhedral product Z_Kn(D¹,S ⁰) arising from the action of the cyclic group C_n and its different subgroups. We use the Riemann-Hurwitz formula to calculate the genus for the quotient spaces involved. There are special maps called Strickland maps: St: Z_K( X,A) → Z_L(X,A) , where X is a commutative topological monoid, and Λ a submonoid. Part 4 proposes a geometric method for calculating the Strickland map in homology in the special case: St: Z_Kn+1(D ¹,S⁰) → Z_Kn(D ¹,S⁰), where D ¹ is a topological monoid under multiplication.

In part 5 we see that, in certain cases, maps on polyhedral products are completely determined by what they do on the maximal full faces of the simplicial complex K. In the last part, we give computational evidence supporting the collapse of the Eilenberg-Moore spectral sequence when computing the cohomology of the space X = U(m) X_T^m Z _K(D²,S ¹)