Localization of Cofibration Categories and Groupoid C∗C^*-algebras

We prove that relative functors out of a cofibration category are essentially the same as relative functors which are only defined on the subcategory of cofibrations. As an application we give a new construction of the functor that assigns to a groupoid its groupoid $C^*$-algebra and thereby its topological $K$-theory spectrum

On the relation between K- and L-theory of complex C*-Algebras

We study the space of natural transformations from connective topological K-theory to algebraic L-theory viewed as functors from C*-algebras to spectra. We prove the existence of a transformation which witnesses the well-known isomorphism between K- and L-groups and becomes an equivalence (of spectra) after inverting 2

Topological 4-manifolds with 4-dimensional fundamental group

Let $\pi$ be a group satisfying the Farrell-Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincar\'e duality space. We consider closed, topological, almost spin manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1 and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel's conjecture, and simply connected manifolds where rigidity is a consequence of Freedman's classification results.Comment: 12 page

A stable ∞\infty-category for equivariant KKKK-theory

For a countable group $G$ we construct a small, idempotent complete, symmetric monoidal, stable $\infty$-category $\mathrm{KK}^{G}_{\mathrm{sep}}$ whose homotopy category recovers the triangulated equivariant Kasparov category of separable $G$-$C^*$-algebras, and exhibit its universal property. Likewise, we consider an associated presentably symmetric monoidal, stable $\infty$-category $\mathrm{KK}^{G}$ which receives a symmetric monoidal functor $\mathrm{kk}^{G}$ from possibly non-separable $G$-$C^*$-algebras and discuss its universal property. In addition to the symmetric monoidal structures, we construct various change-of-group functors relating these KK-categories for varying $G$. We use this to define and establish key properties of a (spectrum valued) equivariant, locally finite $K$-homology theory on proper and locally compact $G$-topological spaces, allowing for coefficients in arbitrary $G$-$C^*$-algebras. Finally, we extend the functor $\mathrm{kk}^{G}$ from $G$-$C^*$-algebras to $G$-$C^*$-categories. These constructions are key in a companion paper about a form of equivariant Paschke duality and assembly maps.Comment: 108 pages. Minor corrections, References update

A survey on operator KK-theory via homotopical algebra

This is a survey article with the goal to advertise spectrum valued versions of $K$- and $KK$- theory for $C^{*}$-algebras via a (stable and symmetric monoidal) $\infty$-categorical enhancement of Kasparov's classical $KK$-theory. The main purpose is to present, in the simplest case, homotopy theoretic arguments for classical results on operator $K$-theory, including Swan's theorems, K\"unneth and universal coefficient formulas, the bootstrap class, variations of Karoubi's conjecture, and spectra of units for strongly self-absorbing $C^*$-algebras, as well as some new aspects on twisted $K$-theory and coherent multiplicative structures on $C^*$-algebras, viewed as objects in the previously mentioned $\infty$-category.Comment: 46 page

Stable classification of 4-manifolds with 3-manifold fundamental groups

We study closed, oriented 4-manifolds whose fundamental group is that of a closed, oriented, aspherical 3-manifold. We show that two such 4-manifolds are stably diffeomorphic if and only if they have the same w2-type and their equivariant intersection forms are stably isometric. We also find explicit algebraic invariants that determine the stable classification for spin manifolds in this class

On the homotopy type of L-spectra of the integers

We show that quadratic and symmetric L-theory of the integers are related by Anderson duality and show that both spectra split integrally into the L-theory of the real numbers and a generalised Eilenberg-Mac Lane spectrum. As a consequence, we obtain a corresponding splitting of the space G/Top. Finally, we prove analogous results for the genuine L-spectra recently devised for the study of Grothendieck--Witt theory.Comment: 32 pages, minor revisions following a referee report, updated reference

Connected sum decompositions of high-dimensional manifolds

The classical Kneser-Milnor theorem says that every closed oriented connected 3-dimensional manifold admits a unique connected sum decomposition into manifolds that cannot be decomposed any further. We discuss to what degree such decompositions exist in higher dimensions and we show that in many settings uniqueness fails in higher dimensions.Comment: 25 pages, fixed several minor mistakes, final versio