arXiv research

In this paper we aim for a generalisation of the Steenrod Approximation Theorem from, concerning a smoothing procedure for sections in smooth locally trivial bundles. The generalisation is that we consider locally trivial smooth bundles with a possibly infinite-dimensional typical fibre. The main result states that a c…

Categorical bundles provide a natural framework for gauge theories involving multiple gauge groups. Unlike the case of traditional bundles there are distinct notions of triviality, and hence also of local triviality, for categorical bundles. We study categorical principal bundles that are product bundles in the categor…

Spaces are classified as almost homology n-manifolds if their homology groups are trivial for all but the last dimension.

We show that there exist non-trivial piecewise-linear (PL) knots with isolated singularities $S^{n-2}\subset S^n$, $n\geq 5$, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally-flat, and topological locally-flat knots, for which it is known that if the complement…

The paper trivializes moment maps for various geometric structures.

New Q-manifolds theory integrates Lie algebroids.

Study local invariants of divergence-free webs in geometry.

New conditions prevent non-trivial relations in local equivalence group.

Extends foliation results to singular cases.

We provide a proof of the controlled surgery sequence, including stability, in the special case that the local fundamental groups are trivial. Stability is a key ingredient in the construction of exotic homology manifolds by Bryant, Ferry, Mio and Weinberger, but no proof has been available. The development given here …

The paper proves a Whitehead theorem for fine shape spaces.

A groupoid is a small category in which each morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all groupoid structure maps are continuous. The notion of monodromy groupoid of a topological groupoid generalises those of fundamental groupoi…

We consider locally conformal Kaehler geometry as an equivariant, homothetic Kaehler geometry (K,Γ). We show that the de Rham class of the Lee form can be naturally identified with the homomorphism projecting Γto its dilation factors, thus completing the description of locally conformal Kaehler geometry in this equivar…

Bundle Networks explore many-to-one maps using fiber bundles and local trivializations.

We recast basic topological concepts underlying differential geometry using the language and tools of noncommutative geometry. This way we characterize principal (free and proper) actions by a density condition in (multiplier) C*-algebras. We introduce the concept of piecewise triviality to adapt the standard notion of…

We consider a diffeological counterpart of the notion of a vector bundle (we call this counterpart a pseudo-bundle, although in the other works it is called differently; among the existing terms there are a "regular vector bundle" of Vincent and "diffeological vector space over X" of Christensen-Wu). The main differenc…

It is proved that an arbitrary finite group acting locally linearly, homologically trivially, and pseudofreely on a closed, simply connected 4-manifold must in fact be cyclic and act semifreely, provided the second betti number of the manifold is at least three.

The Degasperis-Procesi equation's solutions define pseudospherical metrics and can lead to surface collapse.

Floer homology vanishes on certain 3-manifolds formed by knots and their mirrors.

A (smooth) dynamical system with transformation group $\mathbb{T}^n$ is a triple $(A,\mathbb{T}^n,α)$, consisting of a unital locally convex algebra $A$, the $n$-torus $\mathbb{T}^n$ and a group homomorphism $α:\mathbb{T}^n\rightarrow\Aut(A)$, which induces a (smooth) continuous action of $\mathbb{T}^n$ on $A$. In this…

Proves conditions for separating regions in homogeneous spaces without trivial topology.

We study the quasi-convergence equivalence of some families of metrics on locally homogeneous closed 4-manifolds with trivial isotropy group, and identify the dimension of each equivalence class under certain conditions.

The paper constructs homotopically non-trivial spheres in complexified spaces.

We give local criteria for smooth non-embeddablity of Levi-flat manifolds. For this purpose, we pose an analogue of Ueda theory on the neighborhood structure of hypersurfaces in complex manifolds with topologically trivial normal bundles.

New concept of quasi-ribbon surface-links simplifies complex surface-links.

Motivated by some questions in the path integral approach to (topological) gauge theories, we are led to address the following question: given a smooth map from a manifold $M$ to a compact group $G$, is it possible to smoothly `diagonalize' it, i.e.~conjugate it into a map to a maximal torus $T$ of $G$? We analyze the …

The local structure of Finsler metrics of constant flag curvature have been historically mysterious. It is proved that every Matsumoto metric of constant flag curvature on a manifold of dimension n \geq 3 is either Riemannian or locally Minkowskian.

Investigate pseudoconvexity of locally trivial holomorphic ball bundles over compact Riemann surfaces.

Locally conformal SKT structures are introduced and studied on Lie groups and their compact quotients.

Control data constructed for smooth weak deformation retraction of stratified spaces.

We prove a localization formula for group-valued equivariant de Rham cohomology of a compact G-manifold. This formula is a non-trivial generalization of the localization formula of Berline-Vergne and Atiyah-Bott for the usual equivariant de Rham cohomology. As an application, we obtain a version of the Duistermaat-Heck…

We study the construction of quasimorphisms on groups acting on trees introduced by Monod and Shalom, that we call median quasimorphisms, and in particular we fully characterise actions on trees that give rise to non-trivial median quasimorphisms. Roughly speaking, either the action is highly transitive on geodesics, i…

A locally conformally Kaehler (l.c.K.) manifold is a complex manifold admitting a Kaehler covering $\tilde M$, with each deck transformation acting by Kaehler homotheties. A compact l.c.K. manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on $\tilde M$. We prove a structure theorem f…

Following Losik's approach to Gelfand's formal geometry, certain characteristic classes for codimension-one foliations coming from the Gelfand-Fuchs cohomology are considered. Sufficient conditions for non-triviality in terms of dynamical properties of generators of the holonomy groups are found. The non-triviality for…

We summarize recent results initiating spectral analysis on pseudo-Riemannian locally symmetric spaces $Γ\backslash G/H$, beyond the classical setting where $H$ is compact (e.g. theory of automorphic forms for arithmetic $Γ$) or $Γ$ is trivial (e.g. Plancherel-type formula for semisimple symmetric spaces).

In this paper we investigate some connections between Topological Dynamics, the theory of G-Principal Bundles, and the theory of Locally Trivial Groupoids.

Study knotted surfaces in simply-connected 4-manifolds with trivial boundary.

Based in the isomorphism between Lie algebroid cohomology and piecewise smooth cohomology, it is proved that the Rham cohomology of a locally trivial Lie groupoid $G$ on a smooth manifold $M$ is isomorphic to the piecewise Rham cohomology of $G$, in which $G$ and $M$ are manifolds without boundary and $M$ is smoothly t…

The paper revisits a claim about a principal bundle over a contractible base and finds it non-trivial.

The non-existence of non-trivial conformally symmetric manifolds in the three-dimensional Riemannian setting is shown. In Lorentzian signature, a complete local classification is obtained. Furthermore, the isometry classes are examined.

A local classification of locally conformal flat Riemannian Einstein-like four-manifolds as well as a local classification of all locally conformal flat Riemannian four-manifolds for which all Jacobi operators have parallel eigenspaces along every geodesic is given. Non-trivial explicit examples are presented. The prob…

This paper simplifies jet bundles of Lie group-valued functions using linear connections.

We use our new type of bounded locally homeomorphic quasiregular mappings in the unit 3-ball to address long standing problems for such mappings. The construction of such mappings comes from our construction of non-trivial compact 4-dimensional cobordisms $M$ with symmetric boundary components and whose interiors have …

Let $F=(F_1, F_2, ..., F_m): \mathbb{C}^n \to \mathbb{C}^m$ be a polynomial dominant mapping with $n>m$. In this paper we give the relations between the bifurcation set of $F$ and the set of values where $F$ is not M-tame as well as the set of generalized critical values of $F$. We also construct explicitly a proper su…

Around 1960, R. Palais and J. Cerf proved a fundamental result relating spaces of diffeomorphisms and imbeddings of manifolds: If V is a submanifold of M, then the map from Diff(M) to Imb(V,M) that takes f to its restriction to V is locally trivial. We extend this and related results into the context of fibered manifol…

The paper shows how compact Kähler manifolds with a special bundle can be broken down into simpler parts.

Derives formulas for extrinsic Paneitz operator and $Q$-curvature in general dimensions.

We call a quaternionic Kaehler manifold with non-zero scalar curvature, whose quaternionic structure is trivialized by a hypercomplex structure, a hyper-Hermitian quaternionic Kaehler manifold. We prove that every locally symmetric hyper-Hermitian quaternionic Kaehler manifold is locally isometric to the quaternionic p…