Discrete Morse Theory

289 Pages·2019·4.476 MB·English

by  Scoville, Nicholas A.

STUDENT MATHEMATICAL LIBRARY
Volume 90
Discrete
Morse Theory
Nicholas A. Scoville
STUDENT MATHEMATICAL LIBRARY
Volume 90
Discrete
Morse Theory
Nicholas A. Scoville
Editorial Board
SatyanL. Devadoss John Stillwell (Chair)
Rosa Orellana Serge Tabachnikov
2010 Mathematics Subject Classification. Primary 55U05, 58E05, 57Q05,
57Q10.
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Contents
Preface ix
Chapter0. WhatisdiscreteMorsetheory? 1
§0.1. Whatisdiscretetopology? 2
§0.2. WhatisMorsetheory? 9
§0.3. SimplifyingwithdiscreteMorsetheory 13
Chapter1. Simplicialcomplexes 15
§1.1. Basicsofsimplicialcomplexes 15
§1.2. Simplehomotopy 31
Chapter2. DiscreteMorsetheory 41
§2.1. DiscreteMorsefunctions 44
§2.2. Gradientvectorfields 56
§2.3. RandomdiscreteMorsetheory 73
Chapter3. Simplicialhomology 81
§3.1. Linearalgebra 82
§3.2. Bettinumbers 86
§3.3. Invarianceundercollapses 95
Chapter4. MaintheoremsofdiscreteMorsetheory 101
v
vi Contents
§4.1. DiscreteMorseinequalities 101
§4.2. Thecollapsetheorem 111
Chapter5. DiscreteMorsetheoryandpersistenthomology 117
§5.1. PersistencewithdiscreteMorsefunctions 117
§5.2. PersistenthomologyofdiscreteMorsefunctions 134
Chapter6. Booleanfunctionsandevasiveness 149
§6.1. ABooleanfunctiongame 149
§6.2. SimplicialcomplexesareBooleanfunctions 152
§6.3. Quantifyingevasiveness 155
§6.4. DiscreteMorsetheoryandevasiveness 158
Chapter7. TheMorsecomplex 169
§7.1. Twodefinitions 169
§7.2. Rootedforests 177
§7.3. ThepureMorsecomplex 179
Chapter8. Morsehomology 187
§8.1. Gradientvectorfieldsrevisited 188
§8.2. Theflowcomplex 195
§8.3. Equalityofhomology 196
§8.4. Explicitformulaforhomology 199
§8.5. ComputationofBettinumbers 205
Chapter9. ComputationswithdiscreteMorsetheory 209
§9.1. DiscreteMorsefunctionsfrompointdata 209
§9.2. Iteratedcriticalcomplexes 220
Chapter10. StrongdiscreteMorsetheory 233
§10.1. Stronghomotopy 233
§10.2. StrongdiscreteMorsetheory 242
§10.3. SimplicialLusternik-Schnirelmanncategory 249
Bibliography 257
Contents vii
Notationandsymbolindex 265
Index 267

Preface
ThisbookservesasbothanintroductiontodiscreteMorsetheoryand
a general introduction to concepts in topology. I have tried to present
thematerialinawayaccessibletoundergraduateswithnomorethan
acourseinmathematicalproofwriting. Althoughsomebookssuchas
[102,132] include a single chapter on discrete Morse theory, and one
[99] treats both smooth and discrete Morse theory together, no book-
lengthtreatmentisdedicatedsolelytodiscreteMorsetheory. Discrete
Morsetheorydeservesbetter: Itservesasatoolinapplicationsasvaried
as combinatorics [16,41,106,108], probability [57], and biology [136].
Morethanthat,itisfascinatingandbeautifulinitsownright. Discrete
Morse theory is a discrete analogue of the “smooth” Morse theory de-
veloped in Marston Morse’s 1925 paper [124], but it is most popularly
knownviaJohnMilnor[116]. FieldsmedalistStephenSmalewentso
far as to call smooth Morse theory “the single greatest contribution of
American mathematics” [144]. This beauty and utility carries over to
thediscretesetting,asmanyoftheresults,suchastheMorseinequali-
ties,havediscreteanalogues. DiscreteMorsetheorynotonlyistopolog-
icalbutalsoinvolvesideasfromcombinatoricsandlinearalgebra. Yetit
iseasytounderstand,requiringnomorethanfamiliaritywithbasicset
theoryandmathematicalprooftechniques. Thuswefindseveralonline
introductionstodiscreteMorsetheorywrittenbyundergraduates. For
ix