Related papers: Horizontal decompositions, I

We classify the holomorphic parabolic geometries on compact complex manifolds of general type. We accomplish this by bounding the numerical dimension of any smooth projective variety in terms of geometric invariants of the flag variety…

Let $P$ be a set of $n$ points in general position on the plane. A set of closed convex polygons with vertices in $P$, and with pairwise disjoint interiors is called a convex decomposition of $P$ if their union is the convex hull of $P$,…

We prove necessary and sufficient conditions for a smooth surface in a 4-manifold X to be pseudoholomorphic with respect to some almost complex structure on X. This provides a systematic approach to the construction of pseudoholomorphic…

We study smooth proper embeddings of compact orientable surfaces in compact orientable $4$-manifolds and elements in the mapping class group of that surface which are induced by diffeomorphisms of the ambient $4$-manifolds. We call such…

Concerns decompositions of smooth 4-manifolds as the union of two handlebodies, each with handles of index <=2 (``Heegard'' decompositions).Sample result: Two 2-complexes are (up to 2-deformation) dual spines of a Heegard decomposition of…

Given a closed, smooth, connected, orientable $4$-manifold $M$, whose integral homology groups can have $2$-torsion, we determine the homotopy decomposition of the double suspension $\Sigma^2M$ as wedge sums of some elementary…

We study infinitesimal deformations of complete hyperbolic surfaces with boundary and with ideal vertices, possibly decorated with horoballs. ``Admissible'' deformations are the ones that pull all horoballs apart; they form a convex cone of…

We show that any closed oriented 3-manifold can be topologically embedded in some simply-connected closed symplectic 4-manifold, and that it can be made a smooth embedding after one stabilization. As a corollary of the proof we show that…

It is known that for coprime integers $p>q\geq 1$, the lens space $L(p^2,pq-1)$ bounds a rational ball, $B_{p,q}$, arising as the 2-fold branched cover of a (smooth) slice disk in $B^4$ bounding the associated 2-bridge knot. Lekilli and…

We extend the theory of relative trisections of smooth, compact, oriented $4$-manifolds with connected boundary given by Gay and Kirby to include $4$-manifolds with an arbitrary number of boundary components. Additionally, we provide…

We study closed orientable manifolds whose topological complexity is at most 3 and determine their cohomology rings. For some of admissible cohomology rings we are also able to identify corresponding manifolds up to homeomorphism.

Two constructions of contact manifolds are presented: (i) products of S^1 with manifolds admitting a suitable decomposition into two exact symplectic pieces and (ii) fibre connected sums along isotropic circles. Baykur has found a…

Let $M_0$ be a compact and orientable 3-manifold. After capping off spherical boundaries with balls and removing any torus boundaries, we prove that the resulting manifold $M$ contains handlebodies of arbitrary genus such that the closure…

A smooth embedding of a closed $3$-manifold $M$ in $\mathbb{R}^4$ may generically be composed with projection to the fourth coordinate to determine a Morse function on $M$ and hence a Heegaard splitting $M=X\cup_\Sigma Y$. However, starting…

We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of…

We classify all closed non-orientable $\mathbb{P}^2$-irreducible 3-manifolds obtained by identifying the faces of a cube. These turn out to be the closed non-orientable $\mathbb{P}^2$-irreducible 3-manifolds with surface-complexity one. We…

For a closed 4-manifold X and closed 3-manifold M we investigate the smallest integer n (perhaps infinity) such that M embeds in the connected sum of n copies of X. It is proven that any lens space (or homology lens space) embeds…

We show that every smooth closed oriented four-manifold admits a decomposition into two co- dimension zero submanifolds with common boundary. Each of these submanifolds carries a structure of a symplectic manifold with pseudo-convex…

This paper provides a topological method to construct all simply-connected, spin, smooth $6$-manifolds with torsion-free homology using simply-connected, smooth $4$-manifolds as building blocks. We explicitly determine the invariants that…

We prove that if a symplectic 4-manifold $X$ becomes a rational 4-manifold after applying rational blow-down surgery, then the symplectic 4-manifold $X$ is originally rational. That is, a symplectic rational blow-up of a rational symplectic…