Markus Szymik

Brauer groups for commutative S-algebras

We investigate a notion of Azumaya algebras in the context of structured ring spectra and give a definition of Brauer groups. We investigate their Galois theoretic properties, and discuss examples of Azumaya algebras arising from Galois descent and cyclic algebras. We construct examples that are related to topological Hochschild cohomology of group ring spectra and we present a K(n)-local variant of the notion of Brauer groups.

Permutations, power operations, and the center of the category of racks

ABSTRACT Racks and quandles are rich algebraic structures that are strong enough to classify knots. Here we develop several fundamental categorical aspects of the theories of racks and quandles and their relation to the theory of permutations. In particular, we compute the centers of the categories and describe power operations on them, thereby revealing free extra structure that is not apparent from the definitions. This also leads to precise characterizations of these theories in the form of universal properties.

Characteristic cohomotopy classes for families of 4-manifolds

Abstract Families of smooth closed oriented 4-manifolds with a complex spin structure are studied by means of a family version of the Bauer-Furuta invariants. The definition is given in the context of parametrised stable homotopy theory, but an interpretation in terms of characteristic cohomotopy classes on Thom spectra associated to the classifying spaces of complex spin diffeomorphism groups is given as well. The theory is illustrated with families of K3 surfaces and mapping tori of diffeomorphisms. It is also related to equivariant invariants.

Homotopies and the universal fixed point property

A topological space has the fixed point property if every continuous self-map of that space has at least one fixed point. We demonstrate that there are serious restraints imposed by the requirement that there be a choice of fixed points that is continuous whenever the self-map varies continuously. To even specify the problem, we introduce the universal fixed point property. Our results apply in particular to the analysis of convex subspaces of Banach spaces, to the topology of finite-dimensional manifolds and CW complexes, and to the combinatorics of Kolmogorov spaces associated with finite posets.

Twisted homological stability for extensions and automorphism groups of free nilpotent groups

We prove twisted homological stability with polynomial coefficients for automorphism groups of free nilpotent groups of any given class. These groups interpolate between two extremes for which homological stability was known before, the general linear groups over the integers and the automorphism groups of free groups. The proof presented here uses a general result that applies to arbitrary extensions of groups, and that has other applications as well.

Quandle cohomology is a Quillen cohomology

We show that the cohomology groups usually associated with racks and quandles agree with the Quillen cohomology groups for the algebraic theories of racks and quandles, respectively. We also explain how this makes available the entire range of tools that comes with a Quillen homology theory, such as long exact sequences (transitivity) and excision isomorphisms (flat base change).

Spectral sequences for Hochschild cohomology and graded centers of derived categories

The Hochschild cohomology of a differential graded algebra, or a differential graded category, admits a natural map to the graded center of its homology category: the characteristic homomorphism. We interpret it as an edge homomorphism in a spectral sequence. This gives a conceptual explanation of the failure of the characteristic homomorphism to be injective or surjective, in general. To illustrate this, we discuss modules over the dual numbers, coherent sheaves over algebraic curves, as well as examples related to free loop spaces and string topology.

Wege und Schleifen

Topologische Probleme sind im Allgemeinen zu kompliziert, um sie direkt zu losen. Es gibt eine ziemlich grobe Methode, sie ubersichtlicher zu gestalten: Sie werden diskretisiert, indem man die Raume durch die Menge ihrer Wegekomponenten ersetzt. Feinere Diskretisierungsmethoden erhalt man dadurch, dass man die Raume erst durch Hilfsraume ersetzt und dann zu den Wegekomponenten ubergeht. Die resultierenden Mengen haben dann oft eine algebraische Struktur, die ihre Bestimmung leichter macht. Dies wird am Ende dieses Kapitels am Beispiel der Kreislinie angedeutet und in den nachfolgenden Kapiteln weiter ausgenutzt.

Zusammenhang und Trennung

In diesem Kapitel werden Eigenschaften topologischer Raume untersucht, die sich von einem Raum auf jeden anderen hierzu homoomorphen Raum ubertragen. Solche Eigenschaften nennt man topologisch. Die wichtigsten sind Zusammenhang, Trennungsaussagen und Kompaktheit. Die ersten beiden werden in diesem Kapitel diskutiert, die letzte dann im nachsten Kapitel.

Kompaktheit und Abbildungsräume

In diesem Abschnitt wird zunachst der Begriff der Kompaktheit topologischer Raume eingefuhrt. Danach wird er zu dem Begriff der eigentlichen Abbildung relativiert. Es folgt ein technischer Anschnitt uber den Satz von Tychonoff, der bei der ersten Lekture ubergangen werden kann. Anschliesend widmen wir uns den Abbildungsraumen: Wie es schon in der Analysis vor allem die Funktionenraume sind, denen das Interesse gilt, so sind es auch in der Topologie die Raume von stetigen Abbildungen, welche auserst wichtige Beispiele von topologischen Raumen liefern. Das Kapitel wird von einem technischen Abschnitt uber die Kategorie der (lokal) kompakt erzeugten Raume abgeschlossen, der zunachst auch ubergangen werden kann.