title: A Leray-Serre spectral sequence for Euclidean buildings
Creators: name: Beljean, Antoine
Publisher: name: University of Muenster, Institutional Repository MIAMI
Year: 2024
Subjects: subject: 514 Topology
subject: 512 Algebra
subject: 516 Geometry

description: Die Leray-Serre Spektralfolge ist ein algebraisches Werkzeug,
das auf Hurewicz-Faserungen angewandt wird. Unter guten Umständen
ermöglicht dieses Werkzeug, eine neue Homologie auf dem Basisraum
zu definieren, eine Homologie mit lokalen Koeffizienten, die außerdem
Informationen über die singuläre Homologie des Totalraumes liefert.
Ein euklidisches Gebäude ist ein besonderer Simplizialkomplex, auf dem
man eine Metrik definieren kann, die aus diesem einen CAT(0)-Raum macht.
Da ein euklidisches Gebäude kontrahierbar ist, ist seine singuläre Homologie
trivial. Also kann man sich eine neue Homologie für euklidische Gebäude wünschen.
In dieser Thesis konstruieren wir eine „Pseudo-Faserung“, in der ein euklidisches
Gebäude die Rolle des Basisraumes spielt, und dazu ein Analogon der Leray-Serre Spektralfolge, das wir auf diese Pseudo-Faserung anwenden.
In einem Schritt dieser Arbeit definieren wir darüber hinaus für bestimmte CAT(0)-Räume
(unter denen euklidische Gebäude sind) ein Analogon des Tangentialbündels riemannscher
Mannigfaltigkeiten, das „Richtungsbündel“ („direction bundle“).
description: The Leray-Serre spectral sequence is an algebraic tool that applies
to Hurewicz fibrations. Under good circumstances, this tool enables
one to define a new homology on the base space of the fibration,
a homology with local coefficients, which will also deliver information
about the singular homology of the total space of the fibration.
Euclidean buildings are particular simplicial complexes that can be
equipped with a metric which makes them into CAT(0) spaces. They often
arise from certain groups that act on them, and they therefore help to
study these groups. Unfortunately, since Euclidean buildings are contractible,
their singular homology is trivial. Therefore, one may wish to obtain a new
homology for these. In this thesis, we construct a 'pseudo-fibration'
in which a Euclidean building fits as the base space, as well as an analogue of
the Leray-Serre spectral sequence that applies to this pseudo-fibration.
At some step of this work, we also define, for certain CAT(0) spaces and
in particular for Euclidean buildings, an analogue of the tangent bundle
of Riemannian manifolds, called the 'direction bundle'.
description: I The direction bundle of a CAT(0) space ..... 23
1 Setting ..... 25
2 The model spaces ..... 28
2.1 κ = 0 : the Euclidean space ..... 28
2.2 κ = 1 : The Sphere ..... 29
2.3 κ = −1 : the hyperbolic space ..... 30
2.4 Summary: the model spaces Mnκ ..... 32
3 The Alexandrov angle ..... 33
4 The κ-cone over a metric space ..... 34
4.1 The κ = 0 case ..... 34
4.2 The κ &lt; 0 case ..... 35
4.3 The κ &gt; 0 case ..... 36
4.4 Examples ..... 36
4.5 Geodesics in Y and geodesics in CκY ..... 37
5 CAT(0)-spaces, CAT(κ)-spaces ..... 38
5.1 The CAT(0) property ..... 38
5.2 A few properties of CAT(0) spaces ..... 40
5.3 Examples of CAT(0) spaces ..... 41
5.4 The CAT(κ) property ..... 44
5.5 Examples of CAT(κ) spaces ..... 47
6 Mκ-simplicial complexes ..... 48
6.1 Simplices, faces, stars in K ..... 48
6.2 The metric on K ..... 49
7 Spherical and Euclidean buildings ..... 55
7.1 Spherical and Euclidean Coxeter complexes ..... 55
7.2 Spherical and Euclidean buildings ..... 74
8 Space of directions ΣxX ..... 86
8.1 Definition of the space of directions ..... 86
8.2 Examples ..... 88
9 The direction bundle of a CAT(0) space ..... 90
9.1 The goal we wish to reach ..... 91
9.2 The construction ..... 92
9.3 The property (DB1) in the case of a connected simply-connected
complete Riemannian manifold with non-positive curvature ..... 97
9.4 The property (DB1) in the case of a uniquely-geodesic
Mκ-simplicial complex with finite shapes (κ ≤ 0) ..... 98
9.5 The property (DB1) in the case of a Euclidean building ..... 100
9.6 Summary about the direction bundle - our first main
result ..... 100
9.7 The equivalence relation ∼ for the CAT(0) space En ..... 100
9.8 The direction bundle of a connected simply-connected
complete Riemannian manifold with non-positive curvature
..... 103
9.9 An important remark about the topology of the direction
bundle ..... 105
II Construction of the spectral sequence ..... 107
10 Spectral sequences - Definitions and properties ..... 111
10.1 Definitions ..... 111
10.2 Filtrations on modules, graded modules and chain complexes
..... 116
10.3 Convergence of a spectral sequence ..... 117
10.4 The spectral sequence induced by a filtration ..... 118
11 Homology with local coefficients ..... 119
11.1 Local coefficient systems ..... 119
11.2 Homology with local coefficients ..... 121
12 The Leray-Serre spectral sequence (for fibrations) ..... 122
13 Construction of the Leray-Serre spectral sequence for Euclidean
buildings ..... 125
13.1 Reminders, notations and new definitions ..... 126
13.2 Replacing the problem with a simplicial version of the
problem ..... 159
13.3 The local coefficient system ..... 170
13.4 The singular chain complex with local coefficients ..... 175
13.5 The filtration on CΔ,L• (|L|) and the associated spectral
sequence ..... 176
13.6 Identifying the first pages of the spectral sequence ..... 179
13.7 Connection with Σ∗X ,→ ΣXqΣ ↠ X ..... 221
13.8 The homology HΔ,sd(K)• (X;H•(X\{∗})) is in general
not trivial ..... 240
13.9 The page E∞•,• ..... 268
13.10 A word about the conjecture of the second Leray-Serre
Theorem for Euclidean buildings of higher dimensions ..... 276