A non-commutative model for higher twistedK-theory

Abstract

We develop an operator algebraic model for twisted K‐theory, which includes the most general twistings as a generalized cohomology theory (that is, all those classified by the unit spectrum bgl1(KU)). Our model is based on strongly self‐absorbing C*‐algebras. We compare it with the known homotopy‐theoretic descriptions in the literature, which either use parametrized stable homotopy theory or ∞‐categories. We derive a similar comparison of analytic twisted K‐homology with its topological counterpart based on generalized Thom spectra. Our model also works for twisted versions of localizations of the K‐theory spectrum, like KU[1/n] or KUdouble-struckQ.

Spherical T-duality and the spherical Fourier–Mukai transform

“…Higher twisted K-theory satisfies all of the usual axioms of a twisted cohomology theory, including crucially the Mayer-Vietoris property as was shown in Theorem 2.7 of Ref. [16].…”

“…To define K-theory on the closed, oriented 7D manifold P , twisted by a closed 7-form H representing k times the generator of H 7 (P, Z), we first recall from Corollary 4.7 in [7] that the generator of H 7 (S 7 , Z) corresponds to the (higher) Dixmier-Douady invariant of an algebra bundle E → S 7 with fibre a stabilized infinite Cuntz C * -algebra O ∞ ⊗ K. Now let f : P → S 7 be a degree k continuous map, then f * (E) → P is an algebra bundle with fibre a stabilized infinite Cuntz C * -algebra O ∞ ⊗ K and Dixmier-Douady invariant equal to k times the generator of H 7 (P, Z). Then, by [16], the 7-twisted K-theory is defined as K * (P, H) = K * (C 0 (P, f * (E))), where C 0 (P, f * (E)) denotes continuous sections of f * (E) vanishing at infinity. This shows that K * (P, H) is well defined, although we will not use the explicit construction.…”

Homotopical and operator algebraic twisted K-theory

“…From here it is immediate that the cycle description of higher twisted K -theory given by Pennig in [20] is multiplicative.…”

“…and note that these induce the usual products on associated twisted cohomology theories. Now, in a series of papers [8,9,20], Dadarlat and Pennig constructed a commutative symmetric ring spectrum K ∞ A representing the homotopy type K A , whenever A is strongly selfabsorbing, i.e., when there exists an isomorphism A ⊗ A ∼ = A with particularly good properties. Similarly, the homotopy type of Aut I (K, A) is then represented by an I-commutative cartesian I-monoid Aut s I (A ⊗ K) admitting an Eckmann-Hilton action on K ∞…”

“…which was denoted KU A,mod • in [20]; the left map is given by tensoring with the identity in the remaining factors, while the right hand map depends on a rank one projection in K (confer the proof of [20, Theorem 2.14 (b)]). Unfortunately, it does not seem to be known whether in general the right hand map is a stable equivalence (it is not usually a π * -isomorphism).…”