title: Spaces of positive scalar curvature metrics and parametrised Morse theory
Creators: name: Bantje, Jannes
Publisher: name: University of Muenster, Institutional Repository MIAMI
Year: 2023
Subjects: subject: 510 Mathematics

description: In dieser Arbeit untersuchen wir den Raum der Metriken positiver skalarer Krümmung R^+(M) auf einer kompakten Spin-Mannigfaltigkeit M^d hoher Dimension. Wie in früheren Arbeiten verwenden wir die Indexdifferenz von Hitchin, um eine Abbildung nach K-Theorie zu erhalten, die als Detektor für nicht-triviale Homotopieklassen von R^+(M) dient. Wir zeigen, dass diese Abbildung unter günstigen Umständen (rational) surjektiv ist. Unsere Ergebnisse verbessern den State of the Art in zwei Aspekten: Erstens gelten sie in allen Dimensionen d ≥ 5 und zweitens sind sie gültig für die höhere Indexdifferenz, die in die K-Theorie der Gruppen-C*-Algebra von G= π_1(M) abbildet. Dies wird durch eine vollständige Überarbeitung der Methodik früherer Ergebnisse mit "parametrisierter Morse-Theorie" als neue treibende Kraft erreicht, die eine Erweiterung der Methoden von Madsen–Weiss darstellt, die von Perlmutter in einem Preprint beschrieben wird. Wir erläutern ebenfalls eine Anwendung unserer Methodik auf die Wirkung der Diffeomorphismengruppe auf dem Raum R^+(M) via Pullback. Hier zeigen wir, dass die Wirkungsabbildung durch den unendlichen Schleifenraum des Madsen--Tillmann--Weiss-Spektrums, das mit der tangentialen Struktur θ : BSpin(d) × BG → BO(d) assoziiert ist, faktorisiert wird.
description: In this thesis we study the space of positive scalar curvature metrics R^+(M) on a compact spin manifold M^d of high dimension. As in previous results, we use Hitchin's index difference to obtain a map into K-theory acting as a detector for non-trivial homotopy classes of R^+(M). Under favourable circumstances we prove this map to be (rationally) surjective. Our results improve the state of the art in two ways: firstly they hold in all dimensions d ≥ 5, secondly they are valid for the higher index difference, which maps into the K-theory of the group C*-algebra of G= π_1(M). This is accomplished by rebuilding the machinery of previous results from scratch using "parametrised Morse theory" as new driving force, which is an extension of the methods of Madsen–Weiss laid out by Perlmutter in a preprint. We also give another application of our machinery in the form of a rigidity result for the action of the diffeomorphism group on the space R^+(M) via pullback. Here we show that the action map factors through the infinite loop space of the Madsen--Tillmann--Weiss spectrum associated to the tangential structure θ : BSpin(d) × BG → BO(d).
description: 1. Introduction ..... 1 1.1. Uniqueness of positive scalar curvature metrics ..... 3 1.2. The action of the diffeomorphism group on the space of psc metrics ..... 6 1.3. Statement of results .....7 1.4. Methods and proof overview ..... 9 1.5. Reader’s guide ..... 15 I. Parametrised Morse theory ..... 17 2. Sheaves and Manifolds ..... 19 2.1. Sheaves and their homotopy theory ..... 19 2.2. Sheaves with category structure ..... 23 2.3. Tangential structures and Thom spectra ..... 25 2.4. Sheaves of manifolds ..... 27 3. The homotopy colimit decomposition ..... 35 3.1. Morse vector bundles and Thom spectra ..... 36 3.2. Sheaves of manifolds with Morse data and localisation ..... 40 3.3. A regularisation technique ..... 43 3.4. Homotopy colimit decomposition ..... 50 3.5. Introducing boundaries ..... 66 4. The main theorems of Madsen–Weiss and Perlmutter ..... 69 4.1. Statement of the main identification ..... 70 4.2. First layer of the proof of Perlmutter’s main theorem ..... 72 4.3. Parametrised surgery and added connectivity ..... 73 4.4. Establishing the homotopy fibre sequence ..... 84 4.5. The high-dimensional Madsen–Weiss theorem ..... 98 II. Application to positive scalar curvature ..... 103 5. Operator families and K-theory ..... 105 5.1. Graded C*-algebras and Clifford modules .....105 5.2. Families of C*-linear differential operators ..... 109 5.3. K-Theory spectra ..... 112 6. Index theory for spaces of manifolds and positive scalar curvature ..... 117 6.1. The basic index theoretic setup ..... 117 6.2. Index maps on sheaves of manifolds ..... 119 6.3. The topological index via assembly ..... 127 6.4. The topological index via twisted cycles ..... 130 6.5. The index theorem for spaces of manifolds ..... 132 7. Spaces of positive scalar curvature metrics and the index difference ..... 135 7.1. Spaces of psc metrics ..... 135 7.2. Prescribed metrics and Gromov–Lawson–Chernysh surgery ..... 137 7.3. Stable metrics ..... 138 7.4. Fibrewise variants ..... 139 7.5. The index difference ..... 140 7.6. The saddle metric ..... 142 8. Homotopy colimit decompositions of psc manifold sheaves ..... 151 8.1. The choice of tangential structure.....152 8.2. Sheaves of manifolds with surgery data and psc metrics ..... 153 8.3. The fibration theorem ..... 154 8.4. The map into the space of positive scalar curvature metrics ..... 157 8.5. Sheaves of manifolds with saddles and product psc metrics ..... 158 8.6. The psc homotopy colimit diagram ..... 161 8.7. The delooped index difference and a comparison theorem ..... 163 9. Factorisation and detection theorems ..... 173 9.1. A factorisation theorem in the closed case ..... 173 9.2. Extending the basic factorisation theorem ..... 176 9.3. Detection results for the space of psc metrics ..... 178 10.The action of the diffeomorphism group ..... 181 10.1. Factorising action maps ..... 181 10.2. Diffeomorphism groups and their action on the space of psc metrics ..... 182 10.3. Classifying spaces of manifold bundles using sheaves ..... 183 10.4. Rigidity results for the action map ..... 186 A. Appendix ..... 191 A.1. Inverse of the Morita equivalence ..... 191 A.2. A short recap of the methods used in earlier work ..... 193 Index ..... 195 Bibliography ..... 197 List of Figures ..... 207