Differential Function Spectra, the Differential Becker-gottlieb Transfer, and Applications to Differential Algebraic K-theory

Description

We develop differential algebraic K-theory for rings of integers in number fields and we construct a cycle map from geometrized bundles of modules over such a ring to the differential algebraic K-theory. We also treat some of the foundational aspects of differential cohomology, including differential function spectra and the differential Becker-Gottlieb transfer. We then state a transfer index conjecture about the equality of the Becker-Gottlieb transfer and the analytic transfer defined by Lott. In support of this conjecture, we derive some non-trivial consequences which are provable by independent means. Cover Title page Chapter 1. Introduction Chapter 2. Differential function spectra 2.1. Differential cohomology –the axioms 2.2. The construction of the differential function spectrum 2.3. Homotopy groups and long exact sequences 2.4. Differential Data and Transformations Chapter 3. Cycle maps 3.1. Introduction 3.2. Complex 𝐾-theory –a warm-up 3.3. The spectrum 𝐾𝑅 3.4. The topological cycle map 3.5. Kamber-Tondeur forms 3.6. Borel’s regulator 3.7. Characteristic forms 3.8. The cycle map for geometric locally constant sheaves 3.9. Some calculations with ̂𝐾𝑅⁰(*) 3.10. Calculation of ̂𝐾𝑅⁰(*) 3.11. The action of \Aut(𝐾𝑅,𝐴,𝑐). 3.12. Determinants 3.13. Rescaling the metric 3.14. Extension of the cycle maps from projective to finitely generated 𝑅-bundles Chapter 4. Transfers in differential cohomology 4.1. Introduction 4.2. Differential Becker-Gottlieb transfer 4.3. Geometric bundles and integration of forms 4.4. Transfer structures and the Becker-Gottlieb transfer 4.5. The left square in (4.8) and the construction of \tr̂ 4.6. Proof of (4.6) 4.7. Functoriality of the transfer for iterated bundles Chapter 5. A transfer index conjecture 5.1. Introduction 5.2. The statement of the transfer index conjecture 5.3. The analytic index 5.4. Discussion of the transfer index conjecture 5.5. Discussion of Lott’s relation Chapter 6. Technicalities 6.1. Categories with weak equivalences and ∞-categories 6.2. Commutative algebras and monoids 6.3. Smooth objects 6.4. Homotopy invariance 6.5. The de Rham complex 6.6. Function spectra with proper support 6.7. Thom and Euler forms 6.8. The normalized Borel regulator map 6.9. More normalizations Bibliography Back Cover