Related papers: Non-smoothable surfaces in the 4-sphere

We construct higher Chow cycles of type (2,1) on some families of K3 surfaces with non-symplectic automorphisms of order 3 and prove that our cycles are indecomposable for very general members. The proof is a combination of some…

Let K be a non-trivial knot in the 3-sphere and let Y be the 3-manifold obtained by surgery on K with surgery-coefficient 1. Using tools from gauge theory and symplectic topology, it is shown that the fundamental group of Y admits a…

Suppose an orientation preserving action of a finite group $G$ on the closed surface $\Sigma_g$ of genus $g>1$ extends over the 3-torus $T^3$ for some embedding $\Sigma_g\subset T^3$. Then $|G|\le 12(g-1)$, and this upper bound $12(g-1)$…

We study compact orientable essential surfaces in knot exteriors in the 3-sphere. The genus $g$, the number of boundary components $b$, and the boundary slope $p/q$ are fundamental invariants of an essential surface. The \textit{realization…

In this paper we study isotopy classes of closed connected orientable surfaces in the standard $3$-sphere. Such a surface splits the $3$-sphere into two compact connected submanifolds, and by using their Heegaard splittings, we obtain a…

In arXiv:1008.3825, Totaro gave examples of a K3 surface such that its automorphism group is not commensurable with an arithmetic group, answering a question of Mazur. We give examples of rational surfaces with the same property. Our…

The unknot U in S^4 has non-unique smooth spanning 3-balls up to isotopy fixing U. Equivalently there are properly embedded non-separating 3-balls in S^1xB^3 not properly isotopic to 1xB^3. More generally there exist non-separating…

In this paper we provide a systematic discussion of how to incorporate orientation preserving symmetries into the treatment of Willmore surfaces via the loop group method. In this context we first develop a general treatment of Willmore…

We follow our study of non-K\"ahler complex structures on $R^4$ that we defined in a previous paper. We prove that these complex surfaces do not admit any smooth complex compactification. Moreover, we give an explicit description of their…

We show that the problem of determining whether a knot in the 3-sphere is non-trivial lies in NP. This is a consequence of the following more general result. The problem of determining whether the Thurston norm of a second homology class in…

This is a preliminary note on a family of minimal surfaces in the 3-sphere defined by a compatible fourth order equation. The minimal surfaces are geometrically characterized either by having a surface of revolution like induced metric, or…

We prove that the moduli spaces of K3 surfaces with non-symplectic involutions are unirational. As a by-product we describe configuration spaces of 4<d<9 points in the projective plane as arithmetic quotients of type IV.

For a 4-manifold $M$ and a knot $k\colon\mathbb{S}^1\hookrightarrow\partial M$ with dual sphere $G\colon\mathbb{S}^2\hookrightarrow\partial M$, we compute the set $\mathbb{D}(M;k)$ of smooth isotopy classes of neat embeddings…

An important difference between high dimensional smooth manifolds and smooth 4-manifolds that in a 4-manifold it is not always possible to represent every middle dimensional homology class with a smoothly embedded sphere. This is true even…

This paper is concerned with the geometry of principal orbits in quaternionic K\"ahler manifolds $M$ of cohomogeneity one. We focus on the complete cohomogeneity one examples obtained from the non-compact quaternionic K\"ahler symmetric…

Using elementary counting methods of weight systems for finite type invariants of knots and integral homology 3-spheres, in the spirit of [B-NG], we answer positively three questions raised in [Ga]. In particular, we exhibit a one-to-one…

Smooth cubic fourfolds are linked to K3 surfaces via their Hodge structures, due to work of Hassett, and via Kuznetsov's K3 category A. The relation between these two viewpoints has recently been elucidated by Addington and Thomas. In this…

We introduce (co)homology theory for multiple group racks and construct cocycle invariants of compact oriented surfaces in the 3-sphere using their 2-cocycles, where a multiple group rack is a rack consisting of a disjoint union of groups.…

The spanning surface defect uses spanning surfaces of a knot in the $3$-sphere to measure how far a knot is from being alternating. We refine the spanning surface defect and extend the definition to take into account surfaces in the…

We prove that there exist infinitely many topologically slice knots which cannot bound a smooth null-homologous disk in any definite 4-manifold. Furthermore, we show that we can take such knots so that they are linearly independent in the…