There are two important statements regarding the Trautman-Bondi mass at null infinity: one is the positivity, and the other is the Bondi mass loss formula, which are both global in nature. The positivity of the quasi-local mass can potentially lead to a local description at null infinity. This is confirmed for the Vaidya spacetime in this note. We study the Wang-Yau quasi-local mass on surfaces of fixed size at the null infinity of the Vaidya spacetime. The optimal embedding equation is solved explicitly and the quasi-local mass is evaluated in terms of the mass aspect function of the Vaidya spacetime.

A surgery on a knot in 3-sphere is called SU(2)-cyclic if it gives a manifold whose fundamental group has no non-cyclic SU(2) representations. Using holonomy perturbations on the Chern-Simons functional, we prove that the distance of two SU(2)-cyclic surgery coefficients is bounded by the sum of the absolute values of their numerators. This is an analog of Culler-Gordon-Luecke-Shalen's cyclic surgery theorem.

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.

Let S be a compact, orientable surface of hyperbolic type and let ψ be a mapping class of S. The present article shows that there exists a finite cover X of S and a lift ψ_X of ψ to X such that the spectral radius of ψ_X, viewed as an automorphism of H_1(X,C), is greater than one if and only if ψ has infinite order and is not a Dehn multitwist. Among the corollaries of this result is the fact that a compact, irreducible 3-manifold with toroidal or empty boundary and positive simplicial volume admits a finite cover for which the multivariable Alexander polynomial is not identically zero and has Mahler measure strictly greater than one; such a manifold also admits a finite cover whose integral first homology has nontrivial torsion. An independent proof of the main result of this paper was given by A. Hadari [Geom. Topol. 24 (2020), no. 4, 1717–1750; MR4173920]. In that paper, the surface was assumed to have at least one boundary component, but stronger control over the required finite covers was obtained.

We introduce an Alexander polynomial for MOY graphs. For a framed trivalent MOY graph G, we refine the construction and obtain a framed ambient isotopy invariant \Delta(G,c)(t). The invariant \Delta(G,c)(t) satisfies a series of relations, which we call MOY type relations, and conversely these relations determine \Delta(G,c)(t). Using them we provide a graphical definition of the Alexander polynomial of a link. Finally, we discuss some properties and applications of our invariants.