Table Of ContentProgress in Mathematics
335
Urtzi Buijs
Yves Félix
Aniceto Murillo
Daniel Tanré
Lie Models in
Topology
Progress in Mathematics
Volume 335
SeriesEditors
Antoine Chambert-Loir, Université Paris-Diderot, Paris, France
Jiang-Hua Lu, The University of Hong Kong, Hong Kong SAR, China
Michael Ruzhansky, Ghent University, Belgium and Queen Mary
University of London, UK
Yuri Tschinkel, Courant Institute of Mathematical Sciences, New York, USA
Moreinformationaboutthisseriesathttp://www.springer.com/series/4848
Urtzi Buijs • Yves Félix • Aniceto Murillo
Daniel Tanré
Lie Models in Topology
Urtzi Buijs Yves Félix
Departamento de Álgebra, Institut de Recherche in Mathématique
Geometría y Topología et Physique
Universidad de Málaga Louvain-la-Neuve, Belgium
Málaga, Spain
Daniel Tanré
Aniceto Murillo Département de Mathématiques
Departamento de Álgebra, Université de Lille
Geometría y Topología Villeneuve d’Ascq, France
Universidad de Málaga
Málaga, Spain
ISSN 0743-1643 ISSN 2296-505X (electronic)
Progress in Mathematics
ISBN 978-3-030-54429-4 ISBN 978-3-030-54430-0 (eBook)
https://doi.org/10.1007/978-3-030-54430-0
Mathematics Subject Classification (2020): 55P62, 17B70, 17B55, 18N40, 18N50
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Ferran Sunyer i Balaguer (1912{1967) was a self-
taught Catalan mathematician who, in spite of a
seriousphysicaldisability,wasveryactiveinresearch
in classical mathematical analysis, an area in which
he acquired international recognition. His heirs cre-
ated the Fundaci(cid:19)o Ferran Sunyer i Balaguer inside
theInstitutd’EstudisCatalanstohonorthememory
of Ferran Sunyer i Balaguer and to promote mathe-
matical research.
Each year, the Fundaci(cid:19)o Ferran Sunyer i Balaguer
and the Institut d’Estudis Catalans award an in-
ternational research prize for a mathematical mono-
graphofexpositorynature.Theprize-winningmono-
graphsarepublishedinthisseries.Detailsaboutthe
prizeandtheFundaci(cid:19)oFerranSunyeriBalaguercan
be found at
https://ffsb.espais.iec.cat/en/
the-ferran-sunyer-i-balaguer-prize/
This book has been awarded the
Ferran Sunyer i Balaguer 2020 prize.
The members of the scienti(cid:12)c commitee
of the 2020 prize were:
Antoine Chambert-Loir
Universit(cid:19)e Paris-Diderot (Paris 7)
Tere M-Seara
Universitat Polit(cid:19)ecnica de Catalunya
Joan Porti
Universitat Aut(cid:18)onoma de Barcelona
Michael Ruzhansky
Imperial College London
Kristian Seip
Norwegian University of Science and Technology
Ferran Sunyer i Balaguer Prize winners
since 2009
2009 Timothy D. Browning
Quantitative Arithmetic of Projective
Varieties, PM 277
2010 Carlo Mantegazza
Lecture Notes on Mean Curvature Flow,
PM 290
2011 Jayce Getz and Mark Goresky
Hilbert Modular Forms with Coe(cid:14)cients in
Intersection Homology and Quadratic Base
Change, PM 298
2012 Angel Cano, Juan Pablo Navarrete and Jos(cid:19)e
Seade
Complex Kleinian Groups, PM 303
2013 Xavier Tolsa
Analytic capacity, the Cauchy transform,
and non-homogeneous Calder(cid:19)on{Zygmund
theory, PM 307
2014 Veronique Fischer and Michael Ruzhansky
Quantization on Nilpotent Lie Groups,
Open Access, PM 314
2015 The scienti(cid:12)c committee decided not to
award the prize
2016 Vladimir Turaev and Alexis Virelizier
Monoidal Categories and Topological Field
Theory, PM 322
2017 Antoine Chambert-Loir, Johannses Nicaise
and Julien Sebag
Motivic Integration, PM 325
2018 Michael Ruzhansky and Durvudkhan
Suragan
Hardy Inequalities on Homogeneous Groups,
PM 327
2019 The scienti(cid:12)c committee decided not to
award the prize
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Background
1.1 Simplicial categories . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.1.1 Simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.1.2 Simplicial complexes . . . . . . . . . . . . . . . . . . . . . . 21
1.1.3 Simplicial chains . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2 Differential categories . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2.1 Commutative differential graded algebras and
the Sullivan model of a space . . . . . . . . . . . . . . . . . 26
1.2.2 Differential graded Lie algebras and
the Quillen model of a space . . . . . . . . . . . . . . . . . 31
1.2.3 Differential graded coalgebras . . . . . . . . . . . . . . . . . 34
1.2.4 Differential graded Lie coalgebras . . . . . . . . . . . . . . . 37
1.2.5 A∞-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.3 Model categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.3.1 Differential model categories . . . . . . . . . . . . . . . . . 48
1.3.2 Cofibrantly generated model categories. . . . . . . . . . . . 50
2 The Quillen Functors L, C and their Duals A, E
2.1 The functors L and C . . . . . . . . . . . . . . . . . . . . . . . . 54
2.2 The functors A and E . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 Complete Differential Graded Lie Algebras
3.1 Complete differential graded Lie algebras . . . . . . . . . . . . . . 72
3.2 The completion of free Lie algebras . . . . . . . . . . . . . . . . . 76
3.3 Completion vs profinite completion. . . . . . . . . . . . . . . . . . 86
4 Maurer–Cartan Elements and the Deligne Groupoid
4.1 Maurer–Cartanelements . . . . . . . . . . . . . . . . . . . . . . . 94
4.2 Exponential automorphisms and
the Baker–Campbell–Hausdorffproduct . . . . . . . . . . . . . . . 96
4.3 The gauge action and the Deligne groupoid . . . . . . . . . . . . . 100
vii
viii Contents
4.4 Applications to deformation theory. . . . . . . . . . . . . . . . . . 107
4.5 The Goldman–Millson Theorem . . . . . . . . . . . . . . . . . . . 109
5 The Lawrence–Sullivan Interval
5.1 Introducing the Lawrence–Sullivaninterval . . . . . . . . . . . . . 118
5.2 The LS interval as a cylinder . . . . . . . . . . . . . . . . . . . . . 121
5.3 The flow of a differential equation, the gauge action
and the LS interval . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.4 Subdivision of the LS interval and a model of the triangle . . . . . 125
5.5 Paths in a cdgl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6 The Cosimplicial cdgl L•
6.1 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.2 Inductive sequences of models of the standard simplices . . . . . . 134
6.3 Sequences of equivariant models of the standard simplices . . . . . 144
6.4 The cosimplicial cdgl L• . . . . . . . . . . . . . . . . . . . . . . . . 147
6.5 An explicit model for the tetrahedron . . . . . . . . . . . . . . . . 148
6.6 Symmetric MC elements of simplicial complexes . . . . . . . . . . 152
7 The Model and Realization Functors
7.1 Introducing the global model and realization functors.
Adjointness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.2 First features of the global model and realization functors . . . . . 163
7.3 The path components and homotopy groups of (cid:2)L(cid:3) . . . . . . . . . 167
7.4 Homologicalbehaviour of L . . . . . . . . . . . . . . . . . . . . . 172
X
7.5 The Deligne groupoid of the global model . . . . . . . . . . . . . . 177
8 A Model Category for cdgl
8.1 The model category . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.2 Weak equivalences and free extensions . . . . . . . . . . . . . . . . 189
8.3 A path object, a cylinder object and homotopy of morphisms . . . 193
8.4 Minimal models of simplicial sets . . . . . . . . . . . . . . . . . . . 199
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
9 The Global Model Functor via Homotopy Transfer
•
9.1 The Dupont calculus on A (Δ ) . . . . . . . . . . . . . . . . . . 204
PL
9.2 Obtaining L• and LX by transfer. . . . . . . . . . . . . . . . . . . 208
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Contents ix
10 Extracting the Sullivan, Quillen and Neisendorfer Models
from the Global Model
10.1 Connecting the global model with the Sullivan, Quillen
and Neisendorfer models. . . . . . . . . . . . . . . . . . . . . . . . 214
10.2 From the Lie minimal model to the Sullivan model
and vice versa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
10.3 Coformal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
11 The Deligne–Getzler–HinichFunctor MC(cid:129) and Equivalence
of Realizations
11.1 The set of Maurer–Cartanelements as a set of morphisms . . . . . 224
•
11.2 Simplicial contractions of A (Δ ) . . . . . . . . . . . . . . . . . . 228
PL
11.3 The Deligne–Getzler–Hinich ∞-groupoid . . . . . . . . . . . . . . 231
11.4 Equivalence of realizations and Bousfield–Kan completion . . . . . 237
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
12 Examples
12.1 Lie models of 2-dimensional complexes. Surfaces . . . . . . . . . . 245
12.2 Lie models of tori and classifying spaces of right-angled
Artin groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
12.3 Lie model of a product . . . . . . . . . . . . . . . . . . . . . . . . 255
12.4 Mapping spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
12.4.1 Lie models of mapping spaces . . . . . . . . . . . . . . . . . 263
12.4.2 Lie models of pointed mapping spaces . . . . . . . . . . . . 266
12.4.3 Lie models of free loop spaces . . . . . . . . . . . . . . . . . 267
12.4.4 Simplicial enrichment of cdgl and cdga . . . . . . . . . . . 269
12.4.5 Complexes of derivations and homotopy groups
of mapping spaces . . . . . . . . . . . . . . . . . . . . . . . 271
12.5 Homotopy invariants of the realization functor . . . . . . . . . . . 275
12.5.1 Action of π1(cid:2)L(cid:3) on π∗(cid:2)L(cid:3) . . . . . . . . . . . . . . . . . . . 276
12.5.2 The rational homotopy Lie algebra of (cid:2)L(cid:3) . . . . . . . . . . 278
12.5.3 Postnikov decomposition of (cid:2)L(cid:3) . . . . . . . . . . . . . . . . 280
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Notation Index
General notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
