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Sufficient conditions are given for the existence of solutions to the discrete LpMinkowski problem for p-capacity when 0 <p <1and 1 <p <2.

Kronheimer-Mrowka recently proved that the Dehn twist along a 3-sphere in the neck of K3#K3 is not smoothly isotopic to the identity. This provides a new example of self-diffeomorphisms on 4-manifolds that are isotopic to the identity in the topological category but not smoothly so. (The first such examples were given by Ruberman.) In this paper, we use the Pin(2)-equivariant Bauer-Furuta invariant to show that this Dehn twist is not smoothly isotopic to the identity even after a single stabilization (connected summing with the identity map on S2×S2). This gives the first example of exotic phenomena on simply connected smooth 4-manifolds that do not disappear after a single stabilization.

In this paper, we study some algebraic topology aspects of Stringc structures, more precisely, from the perspective of Whitehead tower and the perspective of the loop group of Spinc(n). We also extend the generalized Witten genera constructed for the first time by Chen et al. [J. Differential Geom. 88 (2011), pp. 1–40] to correspond to Stringc structures of various levels and give vanishing results for them.