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Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic $K$-theory

About this Title

Ulrich Bunke, NWF I - Mathematik, Universität Regensburg, 93040 Regensburg, GERMANY and David Gepner, Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907

Publication: Memoirs of the American Mathematical Society
Publication Year: 2021; Volume 269, Number 1316
ISBNs: 978-1-4704-4685-7 (print); 978-1-4704-6470-7 (online)
DOI: https://doi.org/10.1090/memo/1316
Published electronically: April 12, 2021
MSC: Primary 19L50, 57R19

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Differential function spectra
  • 3. Cycle maps
  • 4. Transfers in differential cohomology
  • 5. A transfer index conjecture
  • 6. Technicalities

Abstract

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.

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