CHAPTER 1 Background: Posets, simplicial complexes, and topology This initial chapter reviews some background material on the fundamentals of “poset topology”, which will be our basic context throughout this book. Many of the concepts and results mentioned in the chapter are elementary— such as that of a poset (partially ordered set) and so the corresponding definitions and proofs will often be given only informally, with sources given for further refer- ence. But the review will in particular allow us to establish some further notation, and to provide some initial examples. More experienced readers will of course be able either to skim through this background chapter, or to skip over it entirely. Chapter overview: the viewpoint of “the three levels”. One advantage of using the context of poset topology is that we can start with just the elementary setup of a poset—but then we will be able to obtain conclusions at the two succes- sively deeper levels of simplicial complexes and topological spaces and furthermore we can do so fairly easily, by using just the elementary combinatorics of the original poset. Correspondingly the first three Sections 1.1— 1.3 introduce these levels of anal- ysis. We mainly focus on our usual special case of a poset of subgroups of some fixed group G, so that we will be considering : subgroup posets the associated simplicial complexes, namely subgroup complexes and some basics of algebraic topology, applied to subgroup posets and complexes. And in particular our discussion will emphasize how we make the transition from one level to the next deeper level. The two subsequent Sections 1.4 and 1.5 then continue the theme of transition— by comparing, at these three levels, the corresponding notions of: mappings and G-actions. These interrelations will be fundamental throughout our later analysis. The final Section 1.6 indicates one further such transition between levels—this time in the “reverse” direction, from deeper to more elementary—by introducing: the barycentric subdivision, described in the viewpoint of face posets, along with some further notions for complexes such as joins and links. 9
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