# Schedule of the Workshop "K-theory in topology and non commutative geometry"

## Monday, August 21

10:15 - 10:50 |
Registration & Welcome coffee |

10:50 - 11:00 |
Opening remarks |

11:00 - 12:00 |
Max Karoubi: Algebraic maps between spheres and Bott periodicity |

12:00 - 13:50 |
Lunch break |

13:50 - 14:50 |
Paulo Carrillo Rouse: On geometric obstructions for Fredholm boundary conditions for manifolds with corners via K-theory |

15:00 - 16:00 |
Grigory Garkusha: Algebraic Kasparov K-theory, framed correspondences and stable motivic homotopy theory (after Cuntz and Voevodsky) |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Pierre Julg: Casimir operator and K-theory for real-rank one simple Lie groups |

afterwards |
Reception |

## Tuesday, August 22

09:30 - 10:30 |
Joachim Cuntz: Semigroup C*-algebras and toric varieties |

10:30 - 11:00 |
Group photo and coffee break |

11:00 - 12:00 |
Francesca Arici: Sphere bundles in noncommutative geometry |

12:00 - 13:50 |
Lunch break |

13:50 - 14:50 |
Ralph Kaufmann: K-theoretical and noncommutative methods for topological properties of materials |

15:00 - 16:00 |
Daniel Kasprowski: Shortening binary complexes and commutativity of K-theory with infinite products |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Michael Weiss: The Hatcher-Waldhausen map as a map of spaces with involution |

## Wednesday, August 23

09:30 - 10:30 |
Wolfgang Lück: The Farrell-Jones Conjecture and its applications |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Guoliang Yu: Quantitative operator K-theory and its applications |

12:00 - |
Lunch break, free afternoon |

16:00 - 16:30 |
Tea and cake |

19:00 - |
Conference dinner at the Restaurant Meyer's (Clemens-August-Str. 51a) |

## Thursday, August 24

09:30 - 10:30 |
Ryszard Nest: On group extensions and the Tate symbol on K_{3} |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Kristian Moi: Real topological Hochschild homology |

12:00 - 13:50 |
Lunch break |

13:50 - 14:50 |
Birgit Richter: Towards topological Hochschild homology of Johnson-Wilson spectra |

15:00 - 16:00 |
Sara Arklint: K-theory for non-simple C*-algebras arising from dynamical systems |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Ralf Meyer: Classifying C*-algebras through homological algebra in triangulated categories |

## Friday, August 25

09:30 - 10:30 |
Christian Ausoni: On THH of topological K-theory |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Erik Kjær Pedersen: Some remarks on the surgery exact sequence |

12:00 - |
Lunch break, farewell |

# Abstracts

## Francesca Arici: Sphere bundles in noncommutative geometry

Cuntz-Pimsner algebras are universal C*-algebras associated to a C*-correspondence and they encode dynamical information. In the case of a self Morita equivalence bimodule they can be thought of as total spaces of a noncommutative circle bundle, and the associated six term exact sequence in K-theory can be interpreted as an operator algebraic version of the classical Gysin sequence for circle bundles. In this talk I will review known results in this direction and report on work in progress concerning the construction of higher dimensional noncommutative sphere bundles in terms of Cuntz-Pimsner algebras of sub-product systems.

Based on (ongoing) joint work with G. Landi and J. Kaad.

## Sara Arklint: K-theory for non-simple C*-algebras arising from dynamical systems

The filtered K-theory of a C*-algebra loosely speaking consists of the K-groups and associated maps of all its ideals and subquotients. I will discuss this invariant's (lack of) success as a classifying functor for general purely infinite C*-algebras with finitely many ideals, and compare this to its success for the Cuntz-Krieger algebras which are C*-algebras arising from dynamical systems that are not always purely infinite. The success story includes strong classification and a complete description of the range of the invariant. This is based on joint work with Bentmann-Katsura and Restorff-Ruiz, (and touches on work of Bentmann-Meyer and Meyer-Nest).

## Christian Ausoni: On THH of topological K-theory

Recently, Eva Hoening (Paris 13) generalized a spectral sequence of Morten Brun for computing topological Hochschild homology to the case of ring spectra. In this talk, I will present a computation of the integral THH groups of connective complex topological K-theory and related spectra, based on this spectral sequence.

## Joachim Cuntz: Semigroup C*-algebras and toric varieties

The coordinate ring of a toric variety is the semigroup ring of a finitely generated subsemigroup of Z^{n}. Such semigroups have the interesting feature that their family of constructible ideals is not independent. This is reflected by torsion phenomena in the K-theory of the semigroup C*-algebra. We give a complete formula for the K-theory in the case of subsemigroups of Z^{2}.

## Grigory Garkusha: Algebraic Kasparov K-theory, framed correspondences and stable motivic homotopy theory (after Cuntz and Voevodsky)

Algebraic Kasparov K-theory is a stable motivic homotopy theory for algebras. A major computation here is the computation of the suspension spectrum of an algebra as an explicit Ω-spectrum. It is based on Cuntz's treatment of Kasparov K-theory (resp. bivariant K-theory of locally convex algebras) as well as on further extension of Cuntz's theory to all algebras by Cortinas-Thom. A computation of the suspension spectrum of a smooth algebraic variety by the speaker and Panin is based on the theory of framed correspondences of Voevodsky. In this talk we will show that both computations, based on Cuntz's and Voevodsky's theories respectively, share lots of common properties. They allow to compute explicitly stable motivic homotopy types of spectra.

## Pierre Julg: Casimir operator and K-theory for real-rank one simple Lie groups

The Kasparov γ element for real-rank one simple Lie groups can be realized with irreducible representations on which the Casimir operator vanishes. We explore the geometric interpretation of such representations, involving the L^{2}-harmonic forms on the associated symmetric space and the Cap-Slovak BGG complex on its boundary.

## Max Karoubi: Algebraic maps between spheres and Bott periodicity

At the beginning of his research, J.-L. Loday proved that an algebraic map between an n-dimensional torus and an n-sphere (n > 1) is necessarily homotopic to a constant map. We generalize this result, relaxing the algebraicity condition by a weaker one, using roughly half of the dimension n. These "partially algebraic" maps between algebraic manifolds lead to the construction of Bott elements in both real and complex K-theory. Part of these results is a joint work with Mariusz Wodzicki.

## Daniel Kasprowski: Shortening binary complexes and commutativity of K-theory with infinite products

Grayson gave a model of higher algebraic K-theory using bounded binary acyclic complexes. I will show that in Grayson's model the complexes of length two suffice to generate the whole group. Moreover, I will prove that the comparison map from Nenashev's model for K_{1} to Grayson's model for K_{1} is an isomorphism. It follows that algebraic K-theory of exact categories commutes with infinite products. This is joint work with Christoph Winges.

## Ralph Kaufmann: K-theoretical and noncommutative methods for topological properties of materials

In the past few years there has been an amplification of the use of K-theory and noncommutative geometry in condensed matter physics. In one direction, noncommutative geometry has been used to understand deformations which arise from subjecting the materials to magnetic fields. Another direction is the use of K-theorical invariants to classfy the behaviour or phases of the materials. We will report on work in these directions which are part of two different collaborations.

## Wolfgang Lück: The Farrell-Jones Conjecture and its applications

We give an introduction to the Farrell-Jones Conjecture which aims at the algebraic K- and L-theory of group rings. It is analogous to the Baum-Connes Conjecture about the topological K-theory of reduced group C*-algebras. We report on the substantial progress about the Farrell-Jones Conjecture which was made in the last years, it is meanwhile known for hyperbolic groups, CAT(0)-groups, S-arithmetic groups and lattices in almost connected Lie groups. We give a survey on its applications, for instance to the Novikov Conjecture, the Borel Conjecture and the classification of hyperbolic groups with a sphere of dimension greater or equal to five as boundary, Poincare duality groups, topological rigidity and so on. We will also give an outline of some work in progress concerning Hecke algebras and a stable version of the Cannon Conjecture.

## Ralf Meyer: Classifying C*-algebras through homological algebra in triangulated categories

Bivariant K-theory and its equivariant analogues are triangulated categories. A homological functor on a triangulated category generates homological algebra in it, including notions of projective resolutions. For objects with a projective resolution of length 1, there is a Universal Coefficient Theorem, which gives rise to classification theorems. More recently, I have proved a more general classification theorem for objects with a projective resolution of length 2. This applies, in particular, to all objects in the bootstrap classes in circle-equivariant KK and KK-theory over a finite unique path space, and to graph C*-algebras with arbitrary finite ideal lattice.

This is joint work with R. Bentmann.

## Kristian Moi: Real topological Hochschild homology

Topological Hochschild homology (THH) was introduced by Bökstedt in order to make computations of algebraic K-theory of rings and ring spectra. For rings with anti-involution, Hesselholt and Madsen introduced real algebraic K-theory, which combines hermitian and algebraic K-theory into a single genuine Z/2-equivariant spectrum. I will report on recent joint work with Dotto, Patchkoria and Reeh on real topological Hochschild homology, which is the corresponding equivariant refinement of THH. The talk will focus on basic ideas and computations.

## Ryszard Nest: On group extensions and the Tate symbol on K_{3}

To a group acting on an n-category one can, under simple assumptions, construct a group (n+1)-cocycle. We will give an explicit construction of the example of the 2-cocycle associated to a polarised Hilbert space and sketch a construction of a 3-cocycle associated to "doubly polarised" Hilbert space, an analytic analogue of a 2-Tate space. We will also relate these cocycles to the invariants of algebraic K-theory related to the Tate tame symbol. On the way we will describe the functorial properties of the the Fredholm determinant, the main tool in these constructions.

This is a report on joint work with Jens Kaad and Jesse Wolfson.

## Erik Kjær Pedersen: Some remarks on the surgery exact sequence

## Birgit Richter: Towards topological Hochschild homology of Johnson-Wilson spectra

In joint work with Christian Ausoni we build on McClure's and Staffeldt's identification of topological Hochschild homology of the Adams summand (aka E(1)) and give a description of THH(E(2)) at an odd prime under the assumption that E(2) is a strictly commutative ring spectrum. I'll present some partial results for THH(E(n)) for general n under the same assumption.

## Paulo Carrillo Rouse: On geometric obstructions for Fredholm boundary conditions for manifolds with corners via K-theory

For a manifold with corners there is a homology theory called conormal homology, defined in terms of faces and incidences and whose cycles correspond geometrically to corner's cycles. Its Euler characteristic, χ_{cn} := χ_{0} − χ_{1}, is given by the alternated sum of the number of (open) faces of a given codimension.

In this talk I will report on joint work with Jean-Marie Lescure (Clermont-Ferrand) and explain the following result: For a compact connected manifold with corners X given as a finite product of manifolds with corners of codimension less or equal to three we have that if X satisfies the Fredholm Perturbation property (every elliptic pseudodifferential b-operator on X can be perturbed by a b-regularizing operator so it becomes Fredholm) then the even Euler corner character of X vanishes, i.e. χ_{0}(X) = 0. I will, if time allows it, explain some partial converse results.

The main theorem behind the above result is the computation of the K-theory groups of the algebra of b-compact operators in terms of conormal homology groups. I will spend the main part of the talk explaining this computation.

In many interesting examples, from codimension two, the even Euler corner character does not vanish. I will sketch, with examples, some topological formulas that one can obtain for the obstruction of a given b-elliptic operator to satisfy the Fredholm perturbation property.

## Michael Weiss: The Hatcher-Waldhausen map as a map of spaces with involution

The Hatcher-Waldhausen map relates the space G/O to the K-theory space of the sphere spectrum. I am planning to sketch arguments (using characteristic classes, indices and the like) showing that the Hatcher-Waldhausen map can be promoted to the status of a map respecting involutions, where the involution on the K-theory side is given by a form of duality.

## Guoliang Yu: Quantitative operator K-theory and its applications

In this lecture, I will give an introduction to quantitative operator K-theory and apply it to compute K-theory of operator algebras which naturally arise from geometry. I will discuss applications to asymptotic behavior of positive scalar curvature when the dimension of the manifolds goes to infinity.