Related papers: Embedding obstructions and 4-dimensional thickenin…

We continue to develop an obstruction theory for embedding 2-spheres into 4-manifolds in terms of Whitney towers. The proposed intersection invariants take values in certain graded abelian groups generated by labelled trivalent trees, and…

In the previous paper, we studied obstructions to the existence of complex sections on almost complex manifolds up to cobordism. We determined the obstruction rationally, in terms of the Chern classes. In this paper, we study the torsion…

The van Kampen-Flores theorem states that the $n$-skeleton of a $(2n+2)$-simplex does not embed into $\mathbb{R}^{2n}$. We give two proofs for its generalization to a continuous map from a skeleton of a certain regular CW complex (e.g. a…

We calculate the Seiberg-Witten invariants of branched covers of prime degree, where the branch locus consists of embedded spheres. Aside from the formula itself, our calculations give rise to some new constraints on configurations of…

In this paper we show that every degree 2 homology class of a 2n-dimensional symplectic manifold is represented by an immersed symplectic surface if it has positive symplectic area. Moreover, the symplectic surface can be chosen to be…

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…

We study conditions under which a finite simplicial complex $K$ can be mapped to $\mathbb R^d$ without higher-multiplicity intersections. An almost $r$-embedding is a map $f: K\to \mathbb R^d$ such that the images of any $r$ pairwise…

This article presents a new and more elementary proof of the main Seiberg-Witten-based obstruction to the existence of Einstein metrics on smooth compact 4-manifolds. It also introduces a new smooth manifold invariant which conveniently…

We compute a primary cohomological obstruction to the existence of an equipartition for j mass distributions in R^d by two hyperplanes in the case 2d-3j = 1. The central new result is that such an equipartition always exists if d=6 2^k +2…

(1) We show that if a presentation of the trivial group is "hard to trivialize", in the sense that lots of Tietze moves are necessary to transform it into the trivial presentation, then the associated presentation complex (which is a…

We prove a theorem on equivariant maps implying the following two corollaries: (1) Let N and M be compact orientable n-manifolds with boundaries such that M\subset N, the inclusion M\to N induces an isomorphism in integral cohomology, both…

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…

For each integer $n$ we construct a simply connected $4$-manifold $X$ admitting a smoothly embedded surface $\Sigma$ of self intersection number $n$ such that the complement of the surface has non-trivial fundamental group. This answers a…

We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils…

We propose an open question that seeks to generalise the Four Colour Theorem from two to three dimensions. As an appetiser, we show that 12 instead of four colours are both sufficient and necessary to colour every 2-complex that embeds in a…

We provide integral curvature bounds for compact Riemannian manifolds that allow isometric immersions into a Euclidean space with low codimension in terms of the Betti numbers.

It is well-known that an n-dimensional Poincar\'{e} complex $X^n$, $n \ge 5$, has the homotopy type of a compact topological $n$-manifold if the total surgery obstruction $s(X^n)$ vanishes. The present paper discusses recent attempts to…

Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R^4 can be described as certain 2-morphisms in the 2-category of `2-tangles in 4…

A clear understanding of topology of higher-dimensional objects is important in many branches of both pure and applied mathematics. In this survey we attempt to present some results of higher-dimensional topology in a way which makes clear…

A notion of dual curve for pseudoholomorphic curves in 4--manifolds turns out to be possible only if the notion of almost complex structure structure is slightly generalized. The resulting structure is as easy (perhaps easier) to work with,…