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…

Establishes necessary and sufficient conditions for smooth triviality of Lie subalgebras and Lie ideals, and proves Moser's trick for foliations.

The paper constructs infinitely many $G$-smoothings of a $G$-manifold.

Nilmanifolds are shown to be diffeomorphic to trivial bundles over tori.

Stiefel-Whitney classes of moment-angle manifolds are trivial.

The simplicial volume of oriented closed connected smooth manifolds that admit a non-trivial smooth $S^1$-action vanishes. In the present work we prove a version of this result for the integral foliated simplicial volume of aspherical manifolds: The integral foliated simplicial volume of aspherical oriented closed conn…

We study a smooth analogue of jumping curves of a holomorphic vector bundle, and use Yang-Mills theory over $S ^{2}$ to show that any non-trivial, smooth Hermitian vector bundle $E$ over a smooth simply connected manifold, must have such curves. This is used to give new examples complex manifolds for which a non-tri…

Smooth complex surfaces with triple intersections using differential geometry.

We describe on any finitely generated group G the space of maps G->C which satisfy the parallelogram identity, f(xy)+f(xy^{-1})=2f(x)+2f(y). It is known (but not well-known) that these functions correspond to Zariski-tangent vectors at the trivial character of the character variety of G in SL_2(C). We study the obstruc…

Boundary Dehn twists become trivial after abelianization.

Smoothly isotopic surfaces in a 4-manifold are stable isotopic if trivial in the 2-homology.

New Q-manifolds theory integrates Lie algebroids.

Study immersions and embeddings of manifolds with trivial normal bundles.

This paper deals with certain results on the number of smooth structures on quaternionic projective spaces, obtained through the computation of inertia group and its analogues, which in turn are computed using techniques from stable homotopy theory. We show that the concordance inertia group is trivial in dimension 20,…

We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic …

Anomaly term vanishes for smooth conical spaces, non-trivial for cones over tori.

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

Computes invariant for smooth h-cobordisms families, proving duality and vanishing theorems.

The abstract explores Artin presentations and their connection to 4-manifolds using triangle groups.

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

Smooth surfaces in simply connected 4-manifolds yield groups with non-trivial homology.

New families of embeddings in 4-manifolds, topologically trivial but smoothly non-trivial.

Paper proves every stable 4-sphere has a unique diffeomorphism class.

This article proves that the parity of the number of Klein-bottle leaves in a smooth cooriented taut foliation is invariant under smooth deformations within taut foliations, provided that every Klein-bottle leaf involved in the counting has non-trivial linear holonomy.

Non-trivial Clifford bundle from loop space tangent bundle.

Smooth families of biholomorphisms between strongly pseudoconvex domains are shown to be smooth.

The stable Andrews-Curtis conjecture in combinatorial group theory is the statement that every balanced presentation of the trivial group can be simplified to the trivial form by elementary moves corresponding to "handle-slides" together with "stabilization" moves. Schoenflies conjecture is the statement that the compl…

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…

It is proved that every disconnected surface-link with meridian-based free fundamental group is a trivial (i.e., an unknotted-unlinked) surface-link. This result is a surface-link version of the author's recent announcement result on smooth unknotting of a surface-knot.

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…

Study shows Brakke flow's non-triviality for smooth boundaries in codimension 1.

We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C^4: the smoothing is a natural S^3-bundle over H^3, its holomorphic geometry is determined by the h…

We show that if $M$ and $N$ have the same homotopy type of simply connected closed smooth $m$-manifolds such that the integral and mod-$2$ cohomologies of $M$ vanish in odd degrees, then their homotopy inertia groups are equal. Let $M^{2n}$ be a closed $(n-1)$-connected $2n$-dimensional smooth manifold. We show that, f…

The paper trivializes moment maps for various geometric structures.

Study Khovanov homology of Turaev genus one links, finding a trivial summand.

New solutions found for complex structures on specific manifolds.

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…

New examples of non-smoothable homeomorphisms of 4-manifolds with boundary found.

Paper defines dynamical coherence for flows and proves it under specific conditions.

Cone structures over minimal products can't be calibrated smoothly.

It is proved that the Wedderburn Theorem on finite division rings implies that all knots and links in the smooth 4-dimensional manifolds are trivial.

Seiberg-Witten theory leads to a delicate interplay between Riemannian geometry and smooth topology in dimension four. In particular, the scalar curvature of any metric must satisfy certain non-trivial estimates if the manifold in question has a non-trivial Seiberg-Witten invariant. However, it has recently been discov…

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…

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…

Study on constraints for topological and smooth realizations of line arrangements and configurations.

Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.

Constructs singular Yamabe solutions via equivariant reduction.

We describe a construction of geometric U-folds compatible with a non-trivial extension of the global formulation of four-dimensional extended supergravity on a differentiable spin manifold. The topology of geometric U-folds depends on certain flat fiber bundles which encode how supergravity fields are globally glued t…