• Topology
• Workshop: Hermitian K-theory and trace methods

• Schedule
• Participants

Video recording

Video Recording

Felix Klein Lectures: November 8, 11, 15, 18 (10 am - 12 noon)

Introduced by Bökstedt in the late eighties, topological Hochschild homology is a manifestation of the dual visions of Connes and Waldhausen to extend de Rham cohomology to the noncommutative setting and to replace algebra by higher algebra. In this expanded setting, topological Hochschild homology takes the place of differential forms; the de Rham differential is replaced by an action of the circle group; and de Rham cohomology is replaced by the Tate cohomology of said circle action.

The resulting cohomology theory has had numerous applications to algebraic K-theory and, more recently, to integral p-adic Hodge theory. The goal of these lectures is to give an introduction to this theory and its applications, and to explore the involution on algebraic K-theory and topological Hochschild homology that the presence of duality generates.

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Recent computations yield new non-nilpotent self maps in the stable motivic homotopy category. One of these, called μ₉, leads to a comparison theorem between the η-local motivic sphere spectrum and the spectrum representing Witt groups over the complex numbers. Moreover, these non-nilpotent elements together with a theory of motivic H-∞-spectra lead to the failure of a naive unstable Kahn-Priddy theorem.

Unfortunately there is no talk available due to technical difficulties with our recording.

I will present joint work with Thomas Nikolaus.

Let A be a ring with involution. Then its algebraic K-theory spectrum K^algA has a canonical C₂-action induced by sending a finitely generated projective module to its dual. One can consider the Tate construction of this C₂-action to obtain a spectrum (K^algA)^tC₂. Due to work of Bruce Williams and Michael Weiss, there is a canonical map Ξ: LA → (K^algA)^tC₂ , where LA denotes the symmetric projective L-theory spectrum of A with its involution.

In this talk I want to study this map in the case where A is a complex C-algebra. In this case we can consider the map K^algA → kA, where kA denotes the connective topological K-theory spectrum of A. Conjecturally, this map is an equivalence with finite coefficients. It follows that the map (K^algA)^tC₂ → kA^tC₂ is also conjecturally an equivalence. I will then try to explain that the composite LA → (K^algA)^tC₂ → kA^tC₂ is a 2-completion. This builds on earlier work with Nikolaus in which we produce a natural map τ_A: kA → LA and calculate its effect on homotopy groups which I will talk about in the first part of the talk. The second part of the talk will then be devoted to the study of the above described map.

If time permits I will try to indicate how this could relate to real algebraic K-theory in the sense of Hesselholt-Madsen and Atiyah-Segal completion questions for this spectrum.

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Following Hesselholt and Madsen's development of the so-called "real" (i.e. Z/2-equivariant) version of algebraic K-theory, Dotto developed a theory of real topological Hochschild homology, which obtains a Z/2-equivariant trace map from real K-theory. The input for real THH is a 1-category with duality, and the output is a Z/2-equivariant spectrum - an ∞-categorical object. We present an outline for a project whose goal is to develop a purely ∞-categorical version of real THH.

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This is a report on joint work with Markus Spitzweck and Oliver Röndigs. We use the first Hopf map to solve the homotopy limit problem for K-theory in the stable motivic homotopy category.

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Voevodsky constructed a filtration on the motivic stable homotopy category by measuring how many (de)suspensions with respect to the Tate circle are required to build a given motivic spectrum. The slices (i.e. the associated graded with respect to this filtration) of several motivic spectra (the motivic Eilenberg-MacLane spectrum, algebraic bordism, algebraic K-theory, the sphere spectrum) have been determined in work of Voevodsky, Levine, and others. In joint work with Paul Arne Ostvaer, we compute the slices of hermitian K-theory a.k.a. higher Grothendieck-Witt theory. One application is a quite natural approach to Milnor's conjecture on quadratic forms.

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References:

Oliver Röndigs, Paul Arne Østvær: Slices of hermitian K-theory and Milnor's conjecture on quadratic forms

Oliver Röndigs, Paul Arne Østvær: The multiplicative structure on the graded slices of Hermitian K-theory and Witt-theory

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In the talk we will construct a Grothendieck-Witt space for any stable infinity category with duality. We will show that if we apply our construction to perfect complexes over a commutative ring in which 2 is invertible we recover the classical Grothendieck-Witt space. In the end we will comment on connective real K-theory spectra for such infinity categories.

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I will present recent joint work with Wolfgang Lück, Holger Reich, and John Rognes [arXiv:1607.03557], in which we use assembly maps to study the topological cyclic homology of group algebras. For any finite group G, for any connective ring spectrum A, and for any prime p, we prove that TC(A[G];p) is determined by TC(A[C];p) as C ranges over the cyclic subgroups of G. More precisely, we prove that for any finite group the assembly map with respect to the family of cyclic subgroups induces isomorphisms on all homotopy groups. For infinite groups, we establish pro-isomorphism, split injectivity, and rational injectivity results, as well as counterexamples to injectivity and surjectivity. In particular, for hyperbolic groups and for virtually finitely generated abelian groups, we show that the assembly map with respect to the family of virtually cyclic subgroups is split injective but in general not surjective – in contrast to what happens in algebraic K-theory. I will also mention stronger results for the related functor C of Bökstedt, Hsiang, and Madsen, which we used in [Adv. Math. 304 (2017), 930–1020, arXiv:1504.03674] to study the rational injectivity of the Farrell-Jones assembly map in algebraic K-theory.