Stable and Unstable Homotopy Theory of Higher Geometric Stacks

• Applicants: Professor Dr. David Gepner, Ph.D.; Professor Jeremiah Heller, Ph.D.
• Subject Area: Mathematics
• Term: from 2012 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 220177288

In this project we undertake a study of the stable and unstable homotopy theory of higher geometric stacks over a base, in both the algebraic and topological settings. This is a generalization of motivic homotopy theory in the algebraic setting and of equivariant homotopy theory in the topological setting. We construct the unstable homotopy category as the A1-localization of the ∞-category of sheaves in the Nisnevich topology. The stable homotopy category is then constructed as the stabilization of these ∞-categories with respect to a fixed set of sphere bundles associated to vector bundles on the base stack. We establish foundational results, such as homotopical purity, for these ∞-categories. As an important calculational tool we prove a duality theorem in this setting for smooth and proper maps. In the algebraic setting a particularly important case occurs when the base is the classifying stack of a linearly reductive algebraic group. Our construction then provides an equivariant motivic homotopy theory, which has thus far only been considered for finite groups. We study the endomorphism ring of the equivariant motivic sphere spectrum which in the case of a finite group G is related to the Grothendieck-Witt group of quadratic forms with G-action. While the analogous case of a compact Lie group is already understood in the topological setting, our theory allows for higher group actions such as string groups or loop groups, for example. We formulate a higher tom Dieck splitting conjecture and study the relationship with Waldhausen's A-theory functor.

• DFG Programme: Research Grants