New methods in algebraic K-theory

• Applicant: Professor Dr. Markus Land
• Subject Area: Mathematics
• Term: from 2019 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 424239956

Algebraic K-theory studies rings through their projective modules and geometric objects like schemes through their vector bundles. It is hence closely related to questions in algebra and algebraic geometry. However, the algebraic K-theory of group rings plays an important role in algebraic topology, in the classification of manifolds: It can decide whether the two boundaries of a given h-cobordism are diffeomorphic. This is the basis of surgery theory, a method of systematically studying the classification of manifolds. It was a vision of Waldhausen, that the K-theory of spectral group rings, an object of stable homotopy theory or higher algebra, is closely related to the group of diffeomorphisms of a manifold. This vision is one of the motivations for studying the K-theory of ring spectra.In this project, one the hand one we wish to access further insight to excision phenomena in K-theory and on the other hand aim to connect a variant of algebraic K-theory to questions of geometric topology. In the first part, the aim is thus to understand structural properties of K-theory which are of particular help for concrete new calculations. The basis for the approach we wish to take in this project is a recent result of Tamme and myself, which describes the failure of excision effectively: For every excision context, we construct a map of ring spectra which determines the failure of excision. One of the two ring spectra involved is part of the excision context, the other, however, is new and constructed out of the given excision context. Many known results follow simply from the existence and formal properties of this new ring spectrum. This project is thus about getting more subtle insight into this new ring, and to use this for questions in excision and concrete calculations.In the second part of the project, we want to connect algebraic and hermitian algebraic K-theory to geometric questions. The basis here is the new construction of a genuine C_2 spectrum KR, called real algebraic K-theory, whose underlying spectrum is algebraic K-theory, whose fixed points, hermitian K-theory, are described as an algebraic cobordism category, and whose geometric fixed points are algebraic L-theory. As in Waldhausen's vision, the real algebraic K-theory of ring spectra is expected to be closely related to geometric questions. The new construction of KR allows to study the real algebraic K-theory of such ring spectra, and the goal of the second part of the proposed project is to draw conclusions of the gained knowledge in question of geometric topology.

• DFG Programme: Research Fellowships
• International Connection: Denmark
• Host: Professor Dr. Lars Hesselholt