Related papers: An obstruction to smooth isotopy in dimension 4

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…

Within the Hamiltonian formulation of diffeomorphism invariant theories we address the problem of how to determine and how to reduce diffeomorphisms outside the identity component.

We prove the following result: Let $(X,g_0)$ be a complete, connected 4-manifold with uniformly positive isotropic curvature and with bounded geometry. Then there is a finite collection $\mathcal{F}$ of manifolds of the form $\mathbb{S}^3…

Let $X$ be a smooth simply connected closed 4-manifold with definite intersection form. We show that any automorphism of the intersection form of $X$ is realized by a diffeomorphism of $X \mathbin{\#} S^2 \times S^2$. This extends and…

We deform two-dimensional topological gravity by making use of its gauge theory formulation. The obtained noncommutative gravity model is shown to be invariant under a class of transformations that reduce to standard diffeomorphisms once…

This note constructs a compact, real-analytic, riemannian 4-manifold ({\Sigma}, g) with the properties that: (1) its geodesic flow is completely integrable with smooth but not real-analytic integrals; (2) {\Sigma} is diffeomorphic to $T^2…

In the paper \cite{wall_1}, C.T.C. Wall proved that two smooth closed simply connected 4-manifolds which are homeomorphic are in fact stably diffeomorphic. We prove a similar result which states that two smooth closed 4-manifolds satisfying…

A unified summary is given of the existence theory of Stein manifolds in all dimensions, based on published and pending literature. Eliashberg's characterization of manifolds admitting Stein structures requires an extra delicate hypothesis…

This paper continues the study of decompositions of a smooth 4-manifold into two handlebodies with handles of index $\leq2$. Part I gave existence results in terms of spines and chain complexes over the fundamental group of the ambient…

We present new smoothing techniques for topologically embedded surfaces in smooth 4-manifolds, which give topological isotopy to a smooth surface. As applications, we prove "topological = smooth" results in dimension 4 for certain disks and…

We construct stable minimal hypersurfaces with simple topology in certain compact $4$-manifolds $X$ with boundary, where $X$ embeds into a smooth manifold homeomorphic to $S^4$. For example, if $X$ is equipped with a Riemannian metric $g$…

We prove that a compactly supported homeomorphism of a smooth manifold of dimension greater or equal to 5 can be approximated uniformly by compactly supported diffeomorphisms if and only if it is isotopic to a diffeomorphism. If the given…

The singular set of a generic map $f: M\to F$ of a manifold $M$ of dimension $m\ge 2$ to an oriented surface $F$ is a closed smooth curve $\Sigma(f)$. We study the parity of the number of components of $\Sigma(f)$. The image $f(\Sigma)$ of…

From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our…

Given a finite group action on a smooth manifold, we study the following question: if two equivariant diffeomorphisms are isotopic, must they be equivariantly isotopic? Birman-Hilden and Maclachlan-Harvey proved the answer is "yes" for most…

We consider closed and orientable immersed hypersurfaces of translational manifolds. Given a vector field on such a hypersurface, we define a perturbation of its Gauss map, which allows us to obtain topological invariants for the immersion…

Diffeomorphisms and an internal symmetry (e.g., local Lorentz invariance) are typically regarded as the symmetries of any geometrical gravity theory, including general relativity. In the first-order formalism, diffeomorphisms can be thought…

This article surveys our ongoing project about the relationship between invariants extending the classical Rohlin invariant of homology spheres and those coming from 4-dimensional (Yang-Mills) gauge theory. The main conjecture towards which…

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 develop an obstruction theory for the existence of gauge equivalences in complete differential graded Lie algebras. Specifically, this theory provides a characterization of homotopy equivalences between differential graded algebras…