Homotopical and operator algebraic twisted K-theory

Abstract

Using the framework for multiplicative parametrized homotopy theory introduced in joint work with C. Schlichtkrull, we produce a multiplicative comparison between the homotopical and operator algebraic constructions of twisted K-theory, both in the real and complex case. We also improve several comparison results about twisted K-theory of $$C^*$$ C ∗ -algebras to include multiplicative structures. Our results can also be interpreted in the $$\infty $$ ∞ -categorical setup for parametrized spectra.

Line Bundle Twists for Unitary Bordism are Ghosts

“…Our motivation for the work presented here arose from the results of [HJ20], see also [HS20]. The authors construct point set models for twisted Spin c bordism and twisted K-theory over K(Z, 3) as well as a model for a twisted Atiyah-Bott-Shapiro orientation α ABS : M Spin c K(Z,3) → K K(Z,3) .…”

“…Their construction goes roughly as follows: An A ∞ -ring spectrum R has a space of units GL 1 (R), which deloops to a classifying space BGL 1 (R). A twist is a map ξ : X → BGL 1 (R) from which we can construct a Thom spectrum X ξ , whose homotopy groups are the ξ-twisted homology groups of X.Our motivation for the work presented here arose from the results of [HJ20], see also [HS20]. The authors construct point set models for twisted Spin c bordism and twisted K-theory over K(Z, 3) as well as a model for a twisted Atiyah-Bott-Shapiro orientation α ABS : M Spin c K(Z,3) → K K(Z,3) .…”

Strict algebraic models for rational parametrised spectra II

“…First introduced to study the Becker-Gottlieb transfer [Cla81], parametrised spectra naturally arise throughout much of algebraic topology as the classifying objects of twisted homology and cohomology theories. In applications, parametrised spectra are frequently either invoked explicitly, such as in recent treatments of twisted K-theory [ABG10,HS20] and generalised Thom spectra [ABG18], or else they provide a useful, though often implicit, contextual backdrop; for example the Eilenberg-Moore and Atiyah-Hirzebruch spectral sequences are fundamentally statements about parametrised spectra, as are many other classical results.…”