Related papers: Algorithms in 4-manifold topology

For a given group $G$, we construct an invariant of flat $G$-connections on 4-manifolds from a finite type involutory quasitriangular Hopf $G$-algebra. Hopf $G$-algebras are generalizations of Hopf algebras, equipped with gradings by $G$.…

Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The mapping class group of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The Torelli group of $X$ is the subgroup of the mapping class…

We continue our program initiated in [arXiv:0912.4261] to consider supersymmetric surface operators in a topologically-twisted N=2 pure SU(2) gauge theory, and apply them to the study of four-manifolds and related invariants. Elegant…

We study closed, connected, spin 4-manifolds up to stabilisation by connected sums with copies of $S^2 \times S^2$. For a fixed fundamental group, there are primary, secondary and tertiary obstructions, which together with the signature…

In this paper we prove two results, one semi-historical and the other new. The semi-historical result, which goes back to Thurston and Riley, is that the geometrization theorem implies that there is an algorithm for the homeomorphism…

We show that there exist smooth, simply connected, four-dimensional spin manifolds which do not admit Einstein metrics, but nonetheless satisfy the strict Hitchin-Thorpe inequality. Our construction makes use of the Bauer/Furuta cohomotopy…

The topology of Stein surfaces and contact 3-manifolds is studied by means of handle decompositions. A simple characterization of homeomorphism types of Stein surfaces is obtained --- they correspond to open handlebodies with all handles of…

This is a survey article about the connections between knot theory and four-dimensional topology. Every four-manifold can be represented in terms of a link, by a Kirby diagram. This point of view has led to progress in computing invariants…

We show a rigidity theorem for the Seiberg-Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of…

We construct an invariant of closed ${\rm spin}^c$ 4-manifolds using families of Seiberg-Witten equations. This invariant is formulated as a cohomology class on a certain abstract simplicial complex consisting of embedded surfaces of a…

The goal of this paper is to demonstrate that, at least for nonsimply connected 4-manifolds, the Seiberg-Witten invariant alone does not determine diffeomorphism type within the same homeomorphism type.

A smooth, compact 4-manifold with a Riemannian metric and b^(2+) > 0 has a non-trivial, closed, self-dual 2-form. If the metric is generic, then the zero set of this form is a disjoint union of circles. On the complement of this zero set,…

In this note we study whether specific elements in the second homology of specific simply connected closed $4$-manifolds can be represented by smooth or topologically flat embedded spheres.

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…

For Hamiltonian circle actions on compact, connected, four-dimensional manifolds, we give a generators and relations description for the even part of the equivariant cohomology, as an algebra over the equivariant cohomology of a point. This…

We give a more geometric approach to an algorithm for deciding whether two hyperbolic 3-manifolds are homeomorphic. We also give a more algebraic approach to the homeomorphism problem for geometric, but non-hyperbolic, 3-manifolds.

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…

We study smooth, proper embeddings of noncompact surfaces in 4-manifolds, focusing on exotic planes and annuli, i.e., embeddings pairwise homeomorphic to the standard embeddings of R^2 and R^2-int D^2 in R^4. We encounter two uncountable…

Algorithms that decompose a manifold into simple pieces reveal the geometric and topological structure of the manifold, showing how complicated structures are constructed from simple building blocks. This note describes a way to…

Families of smooth closed oriented 4-manifolds with a complex spin structure are studied by means of a family version of the Bauer--Furuta invariants in the context of parametrised stable homotopy theory, leading to a definition of…