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Geometry-Topology Seminar

Thursday, November 11, 2021 - 5:15pm

Thomas Brazelton



University of Pennsylvania

via Zoom

The Zoom link will be sent via e-mail to all Penn faculty and students. Otherwise, please just e-mail for the link. After the talk, we will stay for a while for a social visit with one another, with the Zoom recording turned off.

Given a finite group G acting on a ring R, Merling constructed an equivariant algebraic K-theory G-spectrum, and work of Malkiewich and Merling, as well as work of Barwick, provides an interpretation of this construction as a spectral Mackey functor. This construction is powerful, but highly categorical; as a result the Mackey functors comprising the homotopy are not obvious from the construction. We will examine the algebraic structure of these homotopy Mackey functors, demonstrating that restriction and transfer data arise as restriction and extension of scalars along twisted group rings. In the case where the group action is trivial, our construction recovers work of Dress and Kuku from the 1980’s which constructs Mackey functors out of the algebraic K-theory of group rings. We develop many families of examples of Mackey functors, both new and old, including K-theory of endomorphism rings, the K-theory of fixed subrings of Galois extensions, and (topological) Hochschild homology of twisted group rings.