Using Seiberg-Witten Floer spectrum and Pin(2)-equivariant KO-theory, we prove new Furuta-type inequalities on the intersection forms of spin cobordisms between homology 3-spheres. As an application, we give explicit constrains on the intersection forms of spin 4-manifolds bounded by Brieskorn spheres ±Σ(2,3,6k±1). Along the way, we also give an alternative proof of Furuta-Kametanni's improvement of 10/8-theorem for closed spin-4 manifolds.

We give a finite presentation for the braid twist group of a decorated surface. If the decorated surface arises from a triangulated marked surface without punctures, we obtain a finite presentation for the spherical twist group of the associated 3-Calabi–Yau triangulated category. The motivation/application is that the result will be used to show that the (principal component of) space of stability conditions on the 3-Calabi–Yau category is simply connected in the sequel [King and Qiu, Invent. Math., to appear].

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.

We propose a finite-dimensional variational principle on triangulated 3-manifolds so that its critical points are related to solutions to Thurstons gluing equation and Hakens normal surface equation. The action functional is the volume.

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