Homotopical and operator algebraic twisted \(K\)-theory (Q2205565)

Twisted \(K\)-theory was first defined from an operator viewpoint, but it also fits into the homotopical framework of twisted cohomology theories. A twisted cohomology theory is represented by a parametrized spectrum over the classifying space \(BGL_1(R)\) of a topological monoid which accounts for the twists. Both real and complex twisted \(K\)-theory are in fact represented by \(E_{\infty}\) parametrized ring spectra and \(BGL_1(KO)\) and \(BGL_1(KU)\) are \(E_{\infty}\) spaces. This paper identifies the operator and homotopical point of view preserving this extra layer of multiplicative structure. The parametrized spectra for both viewpoints of twisted \(K\)-theory are constructed as commutative symmetric ring spectra in retractive spaces. The base spaces, which are the classifying spaces of the topological monoids which account for the twists, are then given by commutative \(\mathcal{I}\)-space monoids, where \(\mathcal{I}\) is the category of finite sets and injections. These two spectra are related through a zigzag of local equivalences. Similar results are obtained for the Spin and Spin\(^c\) bordism spectra, and for twisted \(K\)-theories associated to strongly self-absorbing, purely infinite \(C^*\)-algebras. In particular, taking the infinite Cuntz algebra, the authors obtain that the cycle description of higher twisted \(K\)-theory given by [\textit{U. Pennig}, J. Topol. 9, No. 1, 27--50 (2016; Zbl 1356.46059)] is multiplicative. Another interesting byproduct of the generality of the constructions is a model for the inclusion of the tautological twists of the Thom spectrum of a commutative parametrized ring spectrum.