Algebraic Topology of Finite Topological Spaces and Applications

Jonathan A. Barmak (Autor)

Buch | Softcover

XVII, 170 Seiten

2011 | 2011
Springer Berlin (Verlag)
978-3-642-22002-9 (ISBN)

Infos

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.

Inhalt

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.

Rezensionen

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)

Produkt-Details

• Erscheint lt. Verlag: 24.8.2011
• Reihe/Serie: Lecture Notes in Mathematics
• Zusatzinfo: XVII, 170 p. 35 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Maße: 155 x 235 mm
• Gewicht: 317 g
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Schlagworte: 05-XX • 06-XX • 52-XX • 55-XX • 55-XX; 05-XX; 52-XX ; 06-XX ; 57-XX • 57-XX • Collapses • combinatorics • Finite topological spaces • Homotopy and simple homotopy types • Weak homotopy equivalences
• ISBN-10: 3-642-22002-9 / 3642220029
• ISBN-13: 978-3-642-22002-9 / 9783642220029
• Zustand: Neuware