Unstable eta-periodic and eta-complete motivic homotopy theory

• Applicant: Professor Dr. Tom Bachmann
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 568120925

The aim of this project is to develop useful and novel decompositions of certain motivic spaces, that is, objects of the motivic homotopy category as defined by Morel and Voevodsky. This will lift successful computational techniques from stable to unstable motivic homotopy theory. More specifically, for a motivic space X, we aim to construct an eta-completion X_eta^ and an eta-periodization eta^{-1}X, provide methods for computing these, and show that the periodization and completion interact via a fracture square. This way, X is decomposed into a "classical" part X_eta^ and an "exotic" part eta^{-1}X. Accordingly, problems about X can be decomposed into a part that is approachable by classical methods, and an "exotic" part that, perhaps surprisingly, is also often approachable (by completely different methods). One interesting technical aspect is that the definition of the unstable eta-periodization draws inspiration from the unstable telescopic homotopy theory of Bousfield. This way certain pathologies of more naive definitions can be avoided. A motivic version of the famous Bousfield--Kuhn functor will be constructed and studied; if all goes well this will allow us to identify S^1-stable eta-periodic motivic homotopy theory in terms of the P^1-stable theory. Our new methods will allow new computations in unstable motivic homotopy theory. For example, our new fracture square may in particular be applied to the space BGL_n and should thus shed new light on algebraic vector bundles on affine varieties.

• DFG Programme: Research Grants