Differential KO-theory: constructions, computations, and applications

TL;DR

This paper advances differential KO-theory by developing new constructions, explicit computations, and applications, including a differential Riemann-Roch theorem, spectral sequence analysis, and examples involving higher structures.

Contribution

It introduces novel differential refinements in KO-theory, explicitly identifies spectral sequence differentials, and applies these to geometric and topological problems.

Findings

Constructed a differential refinement of the $$-genus and a pushforward for a Riemann-Roch theorem.

Explicitly identified differentials in the differential Atiyah-Hirzebruch spectral sequence for KO-theory.

Provided applications to higher tangential structures, Adams operations, and a differential Wu formula.

Abstract

We provide several constructions in differential KO-theory. First, we construct a differential refinement of the $\hat{A}$-genus and a pushforward leading to a Riemann-Roch theorem. We set up a differential refinement of the Atiyah-Hirzebruch spectral sequence (AHSS) for differential KO-theory and explicitly identify the differentials, including ones which mix geometric and topological data. We highlight the power of these explicit identifications by providing a characterization of forms in the image of the Pontrjagin character. Along the way, we fill gaps in the literature where K-theory is usually worked out leaving KO-theory essentially untouched. We also illustrate with examples and applications, including higher tangential structures, Adams operations, and a differential Wu formula.