Stable Diffeomorphism Groups of~4-Manifolds

Abstract

Abstract. 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 of Riemann surfaces. In dimension 4 there are many essentially different candidates for the n-manifold Y to choose from. It is shown that the Bauer-Furuta invariants provide invariants in the case Y = CP 2 , which is related to the birational classification of complex surfaces. This will be the case for other Y only after localisation of the target category. In this context, it is shown that the K3-stable Bauer-Furuta invariants determine the S 2 ×S 2 -stable invariants.

Abelian quandles and quandles with abelian structure group

“…In [SV24] the theorem was extended to pq-generated groups G, which allow conjugation and generalised power relations (a k = b m , k, m ∈ Z) in their presentation, and to appropriate quotients of the structure group As(Conj(G)). Those quotients were studied in detail there.…”

The stable homotopy theory of vortices on Riemann surfaces

“…The preceding theory that culminated in Theorem 6.3 has extensions to both the cases where compact Lie groups act on the situation, and where one considers families. This can be set up with only minor changes to the author's previous work in the context of the Bauer-Furuta invariants, see [Szy02], [Szy08], [Szy10], and [Szy12]. Therefore, the form and function of these extensions in the context of the vortex equations will only be sketched here.…”

Bauer-Furuta Invariants and Galois Symmetries

“…This can be related to the family version of the Bauer-Furuta invariants defined in [25], as explained in Section 7 of loc. cit.. See also [24].…”