Algebraic Topology of Finite Topological Spaces and Applications (Lecture Notes in Mathematics, 2032) 2011th Edition by Jonathan A. Barmak (PDF)

Description

This volume deals with the theory of finite topological spaces and its relationship with the homotopy and simple homotopy theory of polyhedra. The interaction between their intrinsic combinatorial and topological structures makes finite spaces a useful tool for studying problems in Topology, Algebra and Geometry from a new perspective. In particular, the methods developed in this manuscript are used to study Quillen’s conjecture on the poset of p-subgroups of a finite group and the Andrews-Curtis conjecture on the 3-deformability of contractible two-dimensional complexes. This self-contained work constitutes the first detailed exposition on the algebraic topology of finite spaces. It is intended for topologists and combinatorialists, but it is also recommended for advanced undergraduate students and graduate students with a modest knowledge of Algebraic Topology.

User’s Reviews

Editorial Reviews: Review From the reviews:“This book deals with the algebraic topology of finite topological spaces and its applications, and includes well-known results on finite spaces and original results developed by the author. The book is self-contained and well written. It is understandable and enjoyable to read. It contains a lot of examples and figures which help the readers to understand the theory.” (Fumihiro Ushitaki, Mathematical Reviews, March, 2014)“This book illustrates convincingly the idea that the study of finite non-Hausdorff spaces from a homotopical point of view is useful in many areas and can even be used to study well-known problems in classical algebraic topology. … This book is a revised version of the PhD Thesis of the author. … All the concepts introduced with the chapters are usefully illustrated by examples and the recollection of all these results gives a very nice introduction to a domain of growing interest.” (Etienne Fieux, Zentralblatt MATH, Vol. 1235, 2012) From the Back Cover This volume deals with the theory of finite topological spaces and itsrelationship with the homotopy and simple homotopy theory of polyhedra.The interaction between their intrinsic combinatorial and topologicalstructures makes finite spaces a useful tool for studying problems inTopology, Algebra and Geometry from a new perspective. In particular,the methods developed in this manuscript are used to study Quillen’sconjecture on the poset of p-subgroups of a finite group and theAndrews-Curtis conjecture on the 3-deformability of contractibletwo-dimensional complexes.This self-contained work constitutes the first detailedexposition on the algebraic topology of finite spaces. It is intendedfor topologists and combinatorialists, but it is also recommended foradvanced undergraduate students and graduate students with a modestknowledge of Algebraic Topology.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Excellent summary of the topology of finite spaces, full of surprising insights. Who would have imagined that finite sets could admit of such rich topological structure?

⭐I used this book heavily in my dissertation research. It really does have everything you could want to know about finite topology. In fact, learning the finite analogs of some concepts in algebraic topology helped me understand general topology more! The pages are also pretty sturdy. I did quite a lot of underlining, highlighting, and note-taking in the margins, and ink never bled through to the opposite page.

⭐It’s not bad, but I was after more general results of finite topological spaces. So in summary, this is not a compilation on finite topological spaces, it is quite focused on the author’s research area

⭐One of the best books I read in the last monts. The author introduce the theory of finite topological space (which is really interesting) and later it’s study from the algebraic topology point of view. This book is a must in every mathematician’s library, it helped me a lot while I was writting my bachelor’s thesis.

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