Combinatorial and Toric Homotopy

The following sections are included:

• CONTENTS
• FOREWORD
• PREFACE

These notes describe some of the homotopy theory surrounding Davis-Januszkiewicz spaces, moment-angle complexes and their generalizations to polyhedral products. These spaces are defined by gluing together prod- ucts formed from pairs of spaces (X, A), where the gluing is determined by the faces of a simplicial complex K. The emphasis is on determining homotopy types — of the spaces themselves, their suspensions and their based loop spaces — and showing how these homotopy types depend on a beautiful interplay between the topology of the pairs (X, A) and the combinatorics of the simplicial complex K.

These lectures are devoted to a remarkable class of 3-dimensional polytopes, which are mathematical models of an important object in quantum physics, quantum chemistry and nanotechnology – fullerenes. The main goal is to show how results of toric topology help to build combinatorial invariants of fullerenes. Fundamental notions are introduced during the lectures. The lecture notes are addressed to a wide audience.

These notes are based on the lectures given by the author at the Young Topologist Seminar, IMS, Singapore (11–21 August 2015).

This text contains an introduction to the theory of limits over the category of presentations, with examples of different well-known functors like homology or derived functors of non-additive functors in the form of derived limits. The theory of so-called fr-codes is also developed. This is a method that shows how different functors from the category of groups to the category of abelian groups, such as group homology and tensor products of abelianization, can be coded as sentences in the alphabet with two symbols f and r.

These notes survey topological problems relevant to the motion planning problem of robotics and include some new results and constructions. First we analyse the notion of topological complexity of configuration spaces which is responsible for discontinuities in algorithms for robot navigation. Then we present explicit motion planning algorithms for coordinated collision free control of many particles moving in Euclidean spaces or on graphs. These algorithms are optimal in the sense that they have minimal number of regions of continuity. Moreover, we describe in full detail the topology of configuration spaces of two particles on a tree and use it to construct some top-dimensional cohomology classes in configuration spaces of n particles on a tree.

The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, Gonzάlez, and Rudyak with the aim of constructing a cellular model of the configuration space of a sphere. Although the original aim was not achieved in the project, the notion of cellular stratified spaces turns out to be useful, at least, in the study of configuration spaces of graphs. In particular, the notion of totally normal cellular stratified spaces was used successfully in a joint work with the former students of the author [1] to study the homotopy type of configuration spaces of graphs with a small number of vertices.