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- MARKUS SZYMIK
- 2007

Families of smooth closed oriented 4-manifolds with a complex spin structure are studied by means of a family version of the Bauer-Furuta in-variants. 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… (More)

- MARKUS SZYMIK
- 2007

Let G be a finite group. For semi-free G-manifolds which are oriented in the sense of Waner [20], the homotopy classes of G-equivariant maps into a G-sphere are described in terms of their degrees, and the degrees occurring are characterised in terms of congruences. This is first shown to be a stable problem, and then solved using methods of equivariant… (More)

- Markus Szymik
- 2011

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 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… (More)

- MARKUS SZYMIK
- 2007

For groups of prime order, equivariant stable maps between equivariant representation spheres are investigated using the Borel co-homology 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)

- Markus Szymik
- 2011

It is shown that the ghost kernel for certain equivariant stable cohomotopy groups of projective spaces is non-trivial. The proof is based on the Borel cohomology Adams spectral sequence and the calculations with the Steen-rod algebra afforded by it.

- Markus Szymik
- 2009

We introduce the notion of a K3 spectrum in analogy with that of an elliptic spectrum and show that there are " enough " K3 spectra in the sense that for all K3 surfaces X in a suitable moduli stack of K3 surfaces there is a K3 spectrum whose underlying ring is isomorphic to the local ring of the moduli stack in X with respect to the etale topology, and… (More)

- Markus Szymik
- 2010

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)

- MARKUS SZYMIK
- 2007

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, 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)

- Markus Szymik, Nathalie Wahl
- 2016

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 V n,r with the homology of the zeroth component of the infinite loop space of the mod n − 1 Moore spectrum. As V = V 2,1 , we can deduce that this group is acyclic. Our proof involves… (More)

- Ehud Meir, Markus Szymik, Joachim Cuntz
- 2015

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 centers: the abelian monoid of isomorphism classes of objects, and the abelian automor-phism group of its identity object. There is an associated obstruction theory that explains… (More)