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  • Book
  • © 2011

Algebraic Topology of Finite Topological Spaces and Applications

  • It is the first complete exposition of the topic

  • Has applications to the study of two long-standing conjectures

  • It is self-contained

  • Includes supplementary material: sn.pub/extras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2032)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xvii
  2. Preliminaries

    • Jonathan A. Barmak
    Pages 1-18
  3. Basic Topological Properties of Finite Spaces

    • Jonathan A. Barmak
    Pages 19-35
  4. Minimal Finite Models

    • Jonathan A. Barmak
    Pages 37-47
  5. Simple Homotopy Types and Finite Spaces

    • Jonathan A. Barmak
    Pages 49-72
  6. Strong Homotopy Types

    • Jonathan A. Barmak
    Pages 73-84
  7. Methods of Reduction

    • Jonathan A. Barmak
    Pages 85-91
  8. h-Regular Complexes and Quotients

    • Jonathan A. Barmak
    Pages 93-104
  9. Group Actions and a Conjecture of Quillen

    • Jonathan A. Barmak
    Pages 105-120
  10. Reduced Lattices

    • Jonathan A. Barmak
    Pages 121-127
  11. Fixed Points and the Lefschetz Number

    • Jonathan A. Barmak
    Pages 129-135
  12. The Andrews–Curtis Conjecture

    • Jonathan A. Barmak
    Pages 137-150
  13. Back Matter

    Pages 151-170

About this book

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.

Keywords

  • 55-XX; 05-XX; 52-XX ; 06-XX ; 57-XX
  • Collapses
  • Finite topological spaces
  • Homotopy and simple homotopy types
  • Weak homotopy equivalences
  • combinatorics

Reviews

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)

Authors and Affiliations

  • Fac. de Cs. Exactas y Naturales, Dept. de Matemática, Universidad de Buenos Aires, Ciudad de Buenos Aires, Argentina

    Jonathan A. Barmak

Bibliographic Information

Buying options

eBook USD 34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 44.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions