Algebraic Topology of Finite Topological Spaces and Applications

Jonathan A. Barmak

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English

Springer-Verlag Berlin and Heidelberg GmbH & Co. K
24 August 2011

Algebra; Analytic geometry; Algebraic geometry; Algebraic topology

Series: Lecture Notes in Mathematics

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.

By: Jonathan A. Barmak
Imprint: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Country of Publication: Germany
Edition: 2011 ed.
Volume: 2032
Dimensions: Height: 235mm, Width: 155mm, Spine: 10mm
Weight: 454g
ISBN: 9783642220029
ISBN 10: 3642220029
Series: Lecture Notes in Mathematics
Pages: 170
Publication Date: 24 August 2011
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

1 Preliminaries.- 2 Basic topological properties of finite spaces.- 3 Minimal finite models.- 4 Simple homotopy types and finite spaces.- 5 Strong homotopy types.- 6 Methods of reduction.- 7 h-regular complexes and quotients.- 8 Group actions and a conjecture of Quillen.- 9 Reduced lattices.- 10 Fixed points and the Lefschetz number.- 11 The Andrews-Curtis conjecture.

Reviews for Algebraic Topology of Finite Topological Spaces and Applications

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)