Legendrian contact homology and nondestabilizability

We completely determine the mod $2$ Seiberg-Witten invariants for any spin structure on any closed, oriented, smooth $4$-manifold $X$. Our computation confirms the validity of the simple type conjecture mod $2$ for spin structures. Our proof also works for families of spin $4$-manifolds and thus computes the mod $2$ Seiberg-Witten invariants for spin families. The proof of our main result uses $Pin(2)$-symmetry to define an enhancement of the mod $2$ Seiberg-Witten invariants. We prove a connected sum formula for the enhanced invariant using localisation in equivariant cohomology. Unlike the usual Seiberg-Witten invariant, the enhanced invariant does not vanish on taking connected sums and by exploiting this property, we are able to compute the enhanced invariant.

math.GT↗

arXiv2026

Schoenflies problem for area-preserving biLipschitz mappings

We prove that any biLipschitz mapping of the boundary of the unit disk onto the boundary of the domain of the same area can be extended to a biLipschitz mapping of the whole plane which preserves the area of any measurable subset.

math.GT↗

arXiv2026

Isotopy versus equivariant isotopy in dimensions three and higher

Given a finite group action on a smooth manifold, we study the following question: if two equivariant diffeomorphisms are isotopic, must they be equivariantly isotopic? Birman-Hilden and Maclachlan-Harvey proved the answer is "yes" for most surfaces. By contrast, we give a general criterion in higher dimensions under which there are many equivariant diffeomorphisms which are isotopic but not equivariantly isotopic. Examples satisfying this criterion include branched covers of split links and "stabilized" branched covers. We prove the result by constructing an invariant valued in the homology of a certain infinite cover of the manifold. We give applications to outer automorphism groups of free products and to group actions on manifolds which fiber over the circle.

math.GT↗