Twists of K-theory and TMF

TL;DR

This paper investigates twisted generalized cohomology theories, especially twisted K-theory and elliptic cohomology, using stable homotopy and quasicategory frameworks, and discusses duality and pushforward maps.

Contribution

It introduces a new approach to twisting elliptic cohomology by degree four classes via stable homotopy and quasicategory theory, extending previous models.

Findings

Twists elliptic cohomology by degree four classes.

Relates twisted cohomology to Postnikov systems.

Discusses Poincare duality and umkehr maps in this context.

Abstract

We explore an approach to twisted generalized cohomology from the point of view of stable homotopy theory and quasicategory theory provided by arXiv:0810.4535. We explain the relationship to the twisted K-theory provided by Fredholm bundles. We show how our approach allows us to twist elliptic cohomology by degree four classes, and more generally by maps to the four-stage Postnikov system BO<0...4>. We also discuss Poincare duality and umkehr maps in this setting.