Related papers: One stabilization is not enough for closed knotted…
In this note, we treat a pair of a K3 surface and a non-symplectic automorphism of order 7m (m=1, 3 and 6) on it. We show that if the fixed locus of a non-symplectic automorphism order 7 is "special" then the pair is unique up to…
We show that a handlebody-knot whose exterior is boundary-irreducible has a unique maximal unnested set of knotted handle decomposing spheres up to isotopies and annulus-moves. As an application, we show that the handlebody-knots $6_{14}$…
We prove that the boundary of the trapped region in an asymptotically Euclidean Riemannian manifold of dimension at least 3 is a stable smooth minimal hypersurface except for a singular set of codimension at least 8.
Conjecturally, a knot is slice if and only if its positive Whitehead double is slice. We consider an analogue of this conjecture for slice disks in the four-ball: two slice disks of a knot are smoothly isotopic if and only if their positive…
We define family versions of the invariant of 4-manifolds with contact boundary due to Kronheimer and Mrowka and use these to detect exotic diffeomorphisms of 4-manifolds with boundary. Further, we show the existence of the first example of…
We study slope stability of smooth surfaces and its connection with exceptional divisors. We show that a surface containing an exceptional divisor with arithmetic genus at least two is slope unstable for some polarisation. In the converse…
Given a nonorientable, locally flatly embedded surface in the $4$-sphere of nonorientable genus $h$, Massey showed that the normal Euler number lies in $\lbrace -2h,-2h+4,\ldots,2h-4,2h \rbrace$. We prove that every such surface with knot…
It is an open problem to provide a characterization of quasiconformally homogeneous Riemann surfaces. We show that given the current literature, this problem can be broken into four open cases with respect to the topology of the underlying…
We develop a $\mathrm{Pin}(2) \times \mathbb{Z}_2$-equivariant refinement of the lattice homotopy type for computing equivariant Seiberg--Witten Floer homotopy types. As an application, we construct a relatively exotic diffeomorphism on a…
We show that, for each integer n, there exist infinitely many pairs of n-framed knots representing homeomorphic but non-diffeomorphic (Stein) 4-manifolds, which are the simplest possible exotic 4-manifolds regarding handlebody structures.…
Four observations compose the main results of this note. The first records the existence of a smoothly embedded 2-sphere $S$ inside $\mathbb{R} P^2\times S^2$ such that performing a Gluck twist on $S$ produces a manifold $Y$ that is…
We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth complex affine quadric of dimension 3 but non algebraically isomorphic to it. We show in particular that…
We address Gromov's band width inequality and Rosenberg's $S^1$-stability conjecture for simply connected smooth four manifolds. Both results are known to be false in dimension 4 due to counterexamples based on Seiberg-Witten invariants.…
We construct examples of non-smoothable surfaces in the $4$-sphere, thereby answering Question 4.32 on the K3 problem list. These surfaces are non-orientable and have knot group of order $2$, thus simultaneously answering Question 4.29(a)…
We prove that oriented and standard shadowing properties are equivalent for topological flows on closed surfaces with the nonwandering set consisting of the finite number of critical elements (i.e., singularities or closed orbits).…
We prove the existence of nonconstant harmonic maps of optimal regularity from an arbitrary closed manifold $(M^n,g)$ of dimension $n>2$ to any closed, non-aspherical manifold $N$ containing no stable minimal two-spheres. In particular,…
Let $f: X \to Y$ be a regular covering of a surface $Y$ of finite type with nonempty boundary, with finitely-generated (possibly infinite) deck group $G$. We give necessary and sufficient conditions for an integral homology class on $X$ to…
For a given branched covering between closed connected surfaces, there are several easy relations one can establish between the Euler characteristics of the surfaces, their orientability, the total degree, and the local degrees at the…
One strategy for distinguishing smooth structures on closed $4$-manifolds is to produce a knot $K$ in $S^3$ that is slice in one smooth filling $W$ of $S^3$ but not slice in some homeomorphic smooth filling $W'$. In this paper we explore…
Recent discoveries in differential topology are reviewed in light of their possible implications for spacetime models and related subjects in theoretical physics. Although not often noted, a particular smoothness (differentiability)…