Trace maps for real algebraic K-theory

• Applicant: Emanuele Dotto, Ph.D.
• Subject Area: Mathematics
• Term: from 2018 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 405348979

The objectives of the project are divided into three categories:1) Geometric properties of real algebraic K-theory,2) Relationship between real K-theory and real topological Hochschild homology (THR),3) Calculations of THR.Each of these categories has two goals.The goals of category 1) complement my current results on the Novikov conjecture, and give a geometric interpretation of the real K-theory of ring spectra. The long term application of these results would provide a conceptual framework, and an interpretation through equivariant homotopy theory, for the relationship between the involution on A-theory and the diffeomorphism groups of manifolds of the fundamental work of Weiss and Williams.1a)[with K.Moi and T.Nikolaus] Describe the Z/2-geometric fixed points of real algebraic K-theory in terms of L-theory.1b)[with K.Moi, H.Reich, M.Varisco] Study the assembly map for real K-theory in Z/2-spectra, via the trace map.The second set of goals aims at establishing a relationship between this geometrically meaningful object KR, and the more computable THR. These goals can potentially be applied to reduce the Novikov conjecture to an approachable statement about real topological cyclic homology.2a) Determine a relationship between real algebraic K-theory and real topological cyclic homology, analogous to the Dundas-Goodwillie-McCarthy Theorem for K-theory.2b) Calculate the Goodwillie tower of Z/2-equivariant calculus for the real K-theory functor, in terms of TR-analogues of THR.The last category addresses calculations. These can eventually give us information about real K-theory, but also establish interesting intrinsic arithmetic properties of real topological and cyclic homology.3a)[with K.Moi and I.Patchkoria] Calculate the real topological Hochschild homology of the integers (at the prime 2), and the trace map from the Hermitian K-theory of the integers.3b)[with K.Moi and I.Patchkoria] Describe the components of the fixed-points of THR by the cyclic groups in terms of a theory of Witt-vectors for Tambara functors. Calculate the Mackey functor of components of TCR(A) for A commutative, and TCR(Z) and TCR(F_p).

• DFG Programme: Priority Programmes
• Subproject of: SPP 1786: Homotopy Theory and Algebraic Geometry
• International Connection: United Kingdom