Algebraic Topology of Finite Topological Spaces and Applications

191 Pages·2011·1.41 MB·English

by  Jonathan A. Barmak

Lecture Notes in Mathematics 2032
Editors:
J.-M. Morel, Cachan
B. Teissier, Paris
For further volumes:
http://www.springer.com/series/304
•
Jonathan A. Barmak
Algebraic Topology
of Finite Topological Spaces
and Applications
123
Jonathan A. Barmak
Departamento de Matema´tica
Fac. Cs. Exactas y Naturales
Universidad de Buenos Aires
Ciudad Universitaria Pabello´n I
(1428) Ciudad de Buenos Aires
Argentina
ISBN 978-3-642-22002-9 e-ISBN 978-3-642-22003-6
DOI 10.1007/978-3-642-22003-6
Springer Heidelberg Dordrecht London New York
Lecture Notes in Mathematics ISSN print edition: 0075-8434
ISSN electronic edition: 1617-9692
Library of Congress Control Number: 2011934806
Mathematics Subject Classification (2010): 55-XX; 05-XX; 52-XX; 06-XX; 57-XX
⃝c Springer-Verlag Berlin Heidelberg 2011
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A las dos bobes
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Preface
There should be more math.
This could be mathier.
B.A. Summers
This book is a revised version of my PhD Thesis [5], supervised by Gabriel
Minian and defended in March 2009 at the Mathematics Department of the
Facultad de Ciencias Exactas y Naturales of the Universidad de Buenos Aires.
Some small changes can be found here, following the suggestions of the ref-
erees and the editors of the LNM.
Gabriel proposed that we work together in the homotopy theory of finite
spaces at the beginning of 2005, claiming that the topic had great potential
and could be rich in applications. Very soon I became convinced of this as well.
A series of notes by Peter May [51–53] and McCord and Stong’s foundational
papers [55, 76] were the starting point of our research. May’s notes contain
very interesting questions and open problems, which motivated the first part
of our work.
This presentation of the theory of finite topological spaces includes the
most fundamental ideas and results previous to our work and, mainly, our
contributions over the last years. It is intended for topologists and combina-
torialists, but since it is a self-contained exposition, it is also recommended
for advanced undergraduate students and graduate students with a modest
knowledge of Algebraic Topology.
The revisions of this book were made during a postdoc at Kungliga
Tekniska ho¨gskolan, in Stockholm.
Acknowledgements
First and foremost, I would like to express my deepest gratitude to Gabriel
Minian, the other person that played a leading role in the creation of this
book, from its beginning as advisor of my undergraduate Thesis, until its
completion with invaluable suggestions on the final presentation. I thank him
for teaching me Algebraic Topology and for sharing with me his expertise.
I gratefully acknowledge his encouragement, guidance and his confidence in
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viii Preface
me. I thank him for developing with me the ideas presented in this book, and
for his unconditional support in each step of my career.
My PhD work at the Universidad de Buenos Aires was supported by a
fellowship of CONICET.
I want to thank Anders Bjo¨rner and the combinatorics group at KTH,
where the revisions took place. My postdoc was supported by grant KAW
2005.0098 from the Knut and Alice Wallenberg Foundation.
I am grateful to the anonymous referees and to the editors of the LNM for
their suggestions that resulted in a better presentation of my work.
Thanks to my family and friends, especially to my parents and brother.
And Mor, and Bubu, of course.
Stockholm, Jonathan A. Barmak
December 2010
Contents
1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Finite Spaces and Posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Maps, Homotopies and Connectedness. . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Homotopy Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Weak Homotopy Types: The Theory of McCord.. . . . . . . . . . . . . . . 10
2 Basic Topological Properties of Finite Spaces . . . . . . . . . . . . . . . . . . . . . 19
2.1 Homotopy and Contiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Minimal Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 T1-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Loops in the Hasse Diagram and the Fundamental Group . . . . . 22
2.5 Euler Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 Automorphism Groups of Finite Posets . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7 Joins, Products, Quotients and Wedges. . . . . . . . . . . . . . . . . . . . . . . . . 29
2.8 A Finite Analogue of the Mapping Cylinder . . . . . . . . . . . . . . . . . . . . 34
3 Minimal Finite Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 A Finite Space Approximation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Minimal Finite Models of the Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Minimal Finite Models of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
∞
3.4 The f (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Simple Homotopy Types and Finite Spaces.. . . . . . . . . . . . . . . . . . . . . . . 49
4.1 Whitehead’s Simple Homotopy Types . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Simple Homotopy Types: The First Main Theorem.. . . . . . . . . . . . 53
4.3 Joins, Products, Wedges and Collapsibility . . . . . . . . . . . . . . . . . . . . . 60
4.4 Simple Homotopy Equivalences: The Second
Main Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 A Simple Homotopy Version of Quillen’s Theorem A.. . . . . . . . . . 68
4.6 Simple, Strong and Weak Homotopy in Two Steps . . . . . . . . . . . . 70
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