Related papers: Counterexamples to the simple loop conjecture in h…

We construct a collection of higher Chow cycles on certain surfaces which degenerate to an arrangement of planes in general position. When its degree is 4, this construction gives a new explicit proof of the Hodge-D-Conjecture for a certain…

Denote by $\Delta_M$ the $M$-dimensional simplex. A map $f\colon \Delta_M\to\mathbb R^d$ is an almost $r$-embedding if $f\sigma_1\cap\ldots\cap f\sigma_r=\emptyset$ whenever $\sigma_1,\ldots,\sigma_r$ are pairwise disjoint faces. A…

Problem 4.19 in Ziegler's "Lectures on Polytopes" asserts that every simple $3$-dimensional polytope has the property that its dual can be constructed as the convex hull of a subset of the vertices of the original simple polytope. In this…

Using the theory of plugs and the self-insertion construction due to the second author, we prove that a foliation of any codimension of any manifold can be modified in a real analytic or piecewise-linear fashion so that all minimal sets…

Work of numerous authors has shown that any smooth, orientable, closed 4-manifold may be described as a loop of Morse functions on a surface, a loop in the cut complex, a loop in the pants complex, or as a multisection. In this paper, we…

For each nonnegative integer m we show that any closed, oriented topological four-manifold with fundamental group Z_{4m+2} and odd intersection form, with possibly seven exceptions, either admits no smooth structure or admits infinitely…

A double disk bundle is any smooth closed manifold obtained as the union of the total spaces of two disk bundles, glued together along their common boundary. The Double Soul Conjecture asserts that a closed simply connected manifold…

We prove that on a closed, orientable surface of genus $g$, a set of simple loops with the property that no two are homotopic or intersect in more than $k$ points has cardinality $\lesssim_k g^{k+1} \log g$. The bound matches the size of…

We prove that, under a simple condition on the cohomology ring, every closed 4-manifold has mod 2 Seiberg-Witten simple type. This result shows that there exists a large class of topological 4-manifolds such that all smooth structures have…

If M is a closed simple 3-manifold whose fundamental group contains a genus-g surface group for some g>1, and if the dimension of H_1(M;Z_2) is at least max(3g-1,6), we show that M contains a closed, incompressible surface of genus at most…

The problem of map enumeration concerns counting connected spatial graphs, with a specified number $j$ of vertices, that can be embedded in a compact surface of genus $g$ in such a way that its complement yields a cellular decomposition of…

We prove the so-called inverse conjecture for the Gowers U^{s+1}-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we establish that if f : [N] -> C is a function with |f(n)| <= 1 for all n and || f…

We discover a simple construction of a four-dimensional family of smooth surfaces of general type with $p_g(S)=q(S)=0$, $K^2_S=3$ with cyclic fundamental group $C_{14}$. We use a degeneration of the surfaces in this family to find…

Any smooth, closed oriented 4-manifold has a surface diagram of arbitrarily high genus g>2 that specifies it up to diffeomorphism. The goal of this paper is to prove the following statement: For any smooth, closed oriented 4-manifold M,…

We present counterexamples to Fujita's conjecture in positive characteristics. Precisely, we show that over any algebraically closed field $k$ of characteristic $p>0$ and for any positive integer $m$, there exists a smooth projective…

An old conjecture of Durfee 1978 bounds the ratio of two basic invariants of complex isolated complete intersection surface singularities: the Milnor number and the singularity (or geometric) genus. We give a counterexample for the case of…

In earlier work, we constructed counterexamples to Lagrangian Poincar\'e recurrence for many toric symplectic four manifolds. Here we provide a few more examples extending the family of counterexamples to include all non-monotone toric…

One can embed arbitrarily many disjoint, non-parallel, non-boundary parallel, incompressible surfaces in any three manifold with at least one boundary component of genus two or greater [4]. This paper proves the contrasting, but not…

We prove the non-rationality of a double cover of $\mathbb{P}^{n}$ branched over a hypersurface $F\subset\mathbb{P}^{n}$ of degree $2n$ having isolated singularities such that $n\ge 4$ and every singular points of the hypersurface $F$ is…

Let $F_g$ be the closed orientable surface of genus $g$. We address the problem to extend torsion elements of the mapping class group ${\mathcal{M}}(F_g)$ over the 4-sphere $S^4$. Let $w_g$ be a torsion element of maximum order in…