Algebraic Topology - A Structural Introduction

372 Pages·2022·7.416 MB·English

by  Marco Grandis

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Other World Scientific Titles by the Author
An Elementary Overview of Mathematical Structures:
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Manifolds and Local Structures: A General Theory
ISBN: 978-981-123-399-9
Category Theory and Applications: A Textbook for Beginners
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ISBN: 978-981-123-608-2
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Published by
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
ALGEBRAIC TOPOLOGY
A Structural Introduction
Copyright © 2022 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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is not required from the publisher.
ISBN 978-981-124-835-1 (hardcover)
ISBN 978-981-124-836-8 (ebook for institutions)
ISBN 978-981-124-837-5 (ebook for individuals)
For any available supplementary material, please visit
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To my colleagues and friends
B1948 Governing Asia
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Preface
AlgebraicTopologyisasystemandstrategyofpartialtranslations,aiming
toreducedifficulttopologicalproblemstoalgebraicfactsthatcanbemore
easily solved. This interface role, between topology and algebra, is the
source of its power and attraction.
Eachofthesetranslationsviewsatopologicalspaceinaparticularlight,
and codifies this picture into an algebraic structure — typically a group,
or a graded abelian group, or a graded ring. In this view, a huge amount
of information is dropped, and a manageable part is kept. Choosing an
appropriate translation, we can hope to solve the topological problem we
are analysing.
Themainsubjectofthisbookissingularhomology,thesimplestofthese
translations. Studying this theory and its applications we also investigate
parts of other disciplines, which form its underlying structural layout (and
in part grew out of it):
- Homological Algebra, for tensor products, the Hom functor and their
derived functors,
- Homotopy Theory, for the homotopy groups and fundamental groupoid,
- Category Theory, the grammar of translations between mathematical
fields.
This book is an introduction to a complex domain, with references to
its advanced parts and ramifications. It is written with minimal prerequi-
sites—basicgeneraltopologyandlittleelse—andamoderateprogression,
startingwithaveryelementarybeginning. Aconsistentpartoftheexposi-
tionisorganisedintheformofexercises, withsuitablehintsandsolutions.
It can be used as a textbook for a first course on Algebraic Topology or
self-study, and a guidebook for further study.
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B1948 Governing Asia
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Contents
Preface page vii
Introduction 1
0.1 Investigating spaces with algebraic structures 1
0.2 Homology and cohomology theories 2
0.3 An outline 3
0.4 Translations and underlying affinities 4
0.5 An inductive approach on structural bases 5
0.6 Prerequisites and literature 5
0.7 Notation and conventions 6
0.8 Acknowledgements 7
1 Introducing Algebraic Topology 9
1.1 Classifying spaces and maps 9
1.2 Transforming problems 17
1.3 The terminology of categories and functors 25
1.4 Paths and homotopy 34
1.5 Natural transformations and equivalence of categories 44
1.6 Products, sums and universal properties 50
2 Singular homology 61
2.1 Chain complexes and their homology 62
2.2 The singular homology groups 72
2.3 Mayer–Vietoris and subdivision 82
2.4 The homology of the spheres 95
2.5 Computing homology 108
2.6 Compact surfaces and projective spaces 115
2.7 Diagram lemmas in Homological Algebra 123
2.8 Complements 127
3 Relative singular homology and homology theories 137
3.1 Main definitions 137
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