22 citations · 79 across the 29 of their papers we have counts for
37 papers
Homological stability fails for the Cremona groups, yet their stable homology is non-trivial
Markus Szymik
The Cremona groups are the groups of all birational transformations of rational varieties, or, in other words, the groups of all automorphisms of rational function fields. These gr…
Artin-Schreier quandles of involutions in absolute Galois groups
Markus Szymik
We introduce a new invariant of fields that refines their real spectrum and is related to their absolute Galois group: the Artin-Schreier quandle. For formally real number fields,…
Generalizations of Loday's assembly maps for Lawvere's algebraic theories
Anna Marie Bohmann, Markus Szymik
Loday's assembly maps approximate the K-theory of group rings by the K-theory of the coefficient ring and the corresponding homology of the group. We present a generalization that…
Groups, conjugation and powers
Markus Szymik, Torstein Vik
We introduce the notion of the power quandle of a group, an algebraic structure that forgets the multiplication but keeps the conjugation and the power maps. Compared with plain qu…
The homotopy types of free racks and quandles
Tyler Lawson, Markus Szymik
We initiate the homotopical study of racks and quandles, two algebraic structures that govern knot theory and related braided structures in algebra and geometry. We prove analogs o…
Preludes to the Eilenberg-Moore and the Leray-Serre spectral sequences
Frank Neumann, Markus Szymik
The Leray-Serre and the Eilenberg-Moore spectral sequences are fundamental tools for computing the cohomology of a group or, more generally, of a space. We describe the relationshi…
Boolean algebras, Morita invariance, and the algebraic K-theory of Lawvere theories
Anna Marie Bohmann, Markus Szymik
The algebraic K-theory of Lawvere theories is a conceptual device to elucidate the stable homology of the symmetry groups of algebraic structures such as the permutation groups and…
The homology of permutation racks
Victoria Lebed, Markus Szymik
Despite a blossoming of research activity on racks and their homology for over two decades, with a record of diverse applications to central parts of contemporary mathematics, ther…
A stable approach to the equivariant Hopf theorem
Markus Szymik
Let G be a finite group. For semi-free G-manifolds which are oriented in the sense of Waner, the homotopy classes of G-equivariant maps into a G-sphere are described in terms of th…
K3 spectra
Markus Szymik
The notion of a K3 spectrum is introduced in analogy with that of an elliptic spectrum and it is shown that there are "enough" K3 spectra in the sense that for all K3 surfaces X in…
The Brauer group of Burnside rings
Markus Szymik
The Brauer group of a commutative ring is an important invariant of a commutative ring, a common journeyman to the group of units and the Picard group. Burnside rings of finite gro…
The chromatic filtration of the Burnside category
Markus Szymik
The Segal map connects the Burnside category of finite groups to the stable homotopy category of their classifying spaces. The chromatic filtrations on the latter can be used to de…
Bauer--Furuta invariants and Galois symmetries
Markus Szymik
The Bauer-Furuta invariants of smooth 4-manifolds are investigated from a functorial point of view. This leads to a definition of equivariant Bauer-Furuta invariants for compact Li…
Characteristic cohomotopy classes for families of 4-manifolds
Markus Szymik
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 in the context of parametrised…
Stable diffeomorphism groups of 4-manifolds
Markus Szymik
A localisation of the category of n-manifolds is introduced by formally inverting the connected sum construction with a chosen n-manifold Y. On the level of automorphism groups, th…
The third Milgram-Priddy class lifts
Markus Szymik
We show that the third cohomology of the finite general linear group with trivial mod 2 coefficients is non-zero. The necessarily unique non-trivial element re…
Spaces of knotted circles and exotic smooth structures
Gregory Arone, Markus Szymik
Suppose that and are closed smooth manifolds of dimension that are homeomorphic. We prove that the spaces of smooth knots and have t…
Adams operations and symmetries of representation categories
Ehud Meir, Markus Szymik
Adams operations are the natural transformations of the representation ring functor on the category of finite groups, and they are one way to describe the usual lambda-ring structu…
Quandle cohomology is a Quillen cohomology
Markus Szymik
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, respectivel…
Alexander-Beck modules detect the unknot
Markus Szymik
We introduce the Alexander-Beck module of a knot as a canonical refinement of the classical Alexander module, and we prove that this new invariant is an unknot-detector.
Permutations, power operations, and the center of the category of racks
Markus Szymik
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 quan…
The rational stable homology of mapping class groups of universal nil-manifolds
Markus Szymik
We compute the rational stable homology of the automorphism groups of free nilpotent groups. These groups interpolate between the general linear groups over the ring of integers an…
Spectral sequences for Hochschild cohomology and graded centers of derived categories
Frank Neumann, Markus Szymik
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 characteristi…
Drinfeld centers for bicategories
Ehud Meir, Markus Szymik
We generalize Drinfeld's notion of the center of a tensor category to bicategories. In this generality, we present a spectral sequence to compute the basic invariants of Drinfeld c…