title: Moduli spaces of proper G-manifolds
Creators: name: Stöveken, Lukas
Publisher: name: University of Muenster, Institutional Repository MIAMI
Year: 2024
Subjects: subject: 510 Mathematics

description: Wir entwickeln semi-simpliziale Techniken für diffeologische Räume und wenden diese an, um zu zeigen, dass Modulräume von eigentlich und äquivariant parametrisierten eigentlichen G-Mannigfaltigkeiten (wobei G eine beliebige Lie Gruppe ist) eine lokal endliche äquivariante Homologietheorie bilden. Darauf aufbauend zeigen wir, dass der Homotopietyp der G-äquivarianten Kobordismuskategorie mit einem unendlichen Schleifenraum eines Spektrums von Modulräumen eigentlicher G-Mannigfaltigkeiten identifiziert werden kann. Darüber hinaus konstruieren wir für jede Lie Gruppe G einen Funktor von G-Räumen nach Spektren, der lokal endlichen G-Bordismus repräsentiert.
description: We introduce semi-simplicial tools for diffeological spaces and apply them to moduli spaces of properly and equivariantly labelled proper G-manifolds (where G is any Lie group) to show that they define a locally finite equivariant homology theory. We use these results to identify the homotopy type of the G-equivariant cobordism category with an infinite loop space associated to a spectrum of moduli spaces. Moreover, for any Lie group G, we find a functor from G-spaces to spectra representing locally finite G-bordism.
description: Introduction ..... 6
I Diffeological spaces ..... 13
1 Basic notions ..... 15
1.1 Pointset diffeology and concrete sheaves ..... 15
1.2 Intermezzo on convenient categories of topological spaces ..... 18
2 Homotopical notions ..... 19
2.1 A model category of diffeological spaces ..... 19
2.2 Dold-Serre fibrations ..... 22
2.3 The Dold-Thom criterion ..... 29
3 Semi-simplicial diffeological spaces ..... 34
3.1 (Semi-)simplicial objects ..... 35
3.2 Diffeological realization of semi-simplicial diffeological spaces ..... 35
3.3 Application to sheaves ..... 42
3.4 Diffeological bar constructions ..... 45
3.5 Classifying spaces of non-unital diffeological categories ..... 48
II Locally finite equivariant homology theories ..... 52
1 Equivariant homology theories ..... 52
1.1 Locally finite equivariant homology theories for one group ..... 53
1.2 Locally finite equivariant homology theories for all groups ..... 58
1.3 Transfers ..... 66
2 The spectral picture ..... 69
2.1 Basics about spectra ..... 69
2.2 Excisive functors ..... 73
2.3 Unstably excisive functors ..... 83
III Moduli spaces of proper equivariantly parametrized manifolds ..... 89
1 Moduli spaces of manifolds ..... 89
1.1 Functoriality ..... 93
1.2 Mainresults ..... 100
1.3 Equivariance ..... 102
2 A diffeological approach to moduli spaces of manifolds ..... 105
2.1 Convenient models ..... 105
2.2 Two stretching lemmas ..... 115
2.3 Relation to locally finite bordism ..... 118
2.4 A weak monoid model ..... 121
2.5 Proof of the main theorem ..... 125
3 Tangential structures and cobordism categories ..... 137
3.1 Tangential structures ..... 137
3.2 Cobordism categories ..... 149
A Some equivariant topology ..... 154
A.1 Basic operations with proper G-spaces ..... 154
A.2 Locally finite G-CW complexes ..... 157
A.3 Equivariantly excisive triads ..... 159
A.4 A G-invariant Ehresmann fibration theorem ..... 161
B Lifting symmetric monoidal functors from simplicial sets to symmetric
spectra ..... 162
B.1 Basics ..... 162
B.2 Limits, colimits and Bousfield-Kan homotopy limits ..... 165
C Elementary diffeological homotopy theory ..... 172
C.1 Simple diffeological spaces ..... 172
C.2 Dold-Serre fibrations and inverse limits ..... 173
Index of Symbols ..... 180
Bibliography ..... 185