Markus Szymik

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For groups of prime order, equivariant stable maps between equivariant representation spheres are investigated using the Borel cohomology Adams spectral sequence. Features of the equivariant stable homotopy category, such as stability and duality, are shown to lift to the category of modules over the associated Steenrod algebra. The dependence on the(More)
We prove that Thompson’s group V is acyclic, answering a 1992 question of Brown in the positive. More generally, we identify the homology of the Higman–Thompson groups Vn,r with the homology of the zeroth component of the infinite loop space of the mod n− 1 Moore spectrum. As V = V2,1, we can deduce that this group is acyclic. Our proof involves(More)
A localisation of the category of n-manifolds is introduced by formally inverting the connected sum construction with a chosen nmanifold Y . On the level of automorphism groups, this leads to the stable diffeomorphism groups of n-manifolds. In dimensions 0 and 2, this is connected to the stable homotopy groups of spheres and the stable mapping class groups(More)
The Bauer-Furuta invariants of smooth 4-manifolds are investigated from a functorial point of view. This leads to a definition of equivariant BauerFuruta invariants for compact Lie group actions. These are studied in Galois covering situations. We show that the ordinary invariants of all quotients are determined by the equivariant invariants of the covering(More)
It is shown that the K3 spectra which refine the local rings of the moduli stack of ordinary p-primitively polarized K3 surfaces in characteristic p allow for an E∞ structure which is unique up to equivalence. This uses the E∞ obstruction theory of Goerss and Hopkins and the description of the deformation theory of such K3 surfaces in terms of their Hodge(More)
A topological space is said to have the fixed point property if every continuous self-map of it has at least one fixed point. In this text, we investigate the additional restraints imposed by the requirement that a fixed point can be chosen continuously when the self-map is varied continuously. While we are able to present a class of universal fixed point(More)
It is shown that the K3 spectra which refine the local rings of the moduli stack of ordinary p-primitively polarized K3 surfaces in characteristic p allow for an E∞ structure which is unique up to equivalence. This uses the E∞ obstruction theory of Goerss and Hopkins and the description of the deformation theory of such K3 surfaces in terms of their Hodge(More)
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 groups play an important rôle in representation theory, and their groups of units and Picard groups have been studied extensively. In this short note, we completely determine the(More)
The purpose of these notes is to show that the methods introduced by Bauer and Furuta, see [5, 6, 7], in order to refine the Seiberg-Witten invariants of smooth 4-dimensional manifolds can also be used to obtain stable homotopy classes from 2-dimensional manifolds, using the vortex equations on the latter. So far these notes contain barely more than the(More)