arXiv research

Spheres in curve complexes are almost simply connected.

Finite rigid sets found in sphere complexes for some but not all cases.

Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.

Research shows arc complex is not quasi-isometric to sphere complex.

By a theorem of Kirchhoff if the six sphere admits an almost complex structure then the seven sphere is parallelizable, more crucial, he exhibited an explicit global frame constructed out of the given almost complex structure. This result implicitly equips the seven sphere with a definite H-space multiplication. We pro…

Researchers classify special curved spheres in a complex space.

We discuss the integrability of orthogonal almost complex structures on Riemannian products of even-dimensional round spheres and give a partial answer to the question raised by E. Calabi concerning the existence of complex structures on a product manifold of a round 2-sphere and a round 4-sphere.

Survey on finite group actions on CW-complexes homotopy to spheres.

Proof of existence of a complex structure on the six-sphere, followed by an explicit computation of its underlying integrable almost complex tensor by the aid of inner automorphisms of the octonions, is exhibited. Both are elementary and self-contained however the size and complexity of the emerging almost complex tens…

In 2003, S.-s. Chern began a study of almost-complex structures on the 6-sphere, with the idea of exploiting the special properties of its well-known almost-complex structure invariant under the exceptional group $G_2$. While he did not solve the (currently still open) problem of determining whether there exists an int…

Study shows a copy of 7D real projective space in the space of almost complex structures on 6-sphere.

We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space.

This article is a survey about or introduction to certain aspects of the complex geometry of a hypothetical complex structure on the six-sphere. We discuss a result of Peternell--Campana--Demailly on the algebraic dimension of a hypothetical complex six-sphere and give some examples. We also give an overview over an ap…

The degree of certain holomorphic 2-spheres is bounded.

Confirming the Powell Conjecture for genus-3 Heegaard splittings of the 3-sphere.

New $f$-vectors reveal geometric Lefschetz-like decompositions of flag spheres.

We show that the disk complex of a genus $g>1$ Heegaard surface for the 3-sphere is homotopy equivalent to a wedge of $(2g-2)$-dimensional spheres. This implies that genus $g>1$ Heegaard surfaces for the 3-sphere are topologically minimal with index $2g-1$.

Curved 10-manifolds with torus symmetry are spheres or complex projective spaces.

Paper confirms Whitehead's conjecture for aspherical 2-complexes.

We introduce and study a new class of homotopy spheres called Farrell-Jones spheres. Using Farrell-Jones sphere we construct examples of closed negatively curved manifolds $M^{2n}$, where $n=7$ or $8$, which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a que…

New tools prove smooth actions on exotic spheres.

Study finds bound on energy of minimal spheres on complex manifolds.

In this paper we study almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres in the space of purely imaginary octaves of the split Cayley algebra $\mathbf{Ca}'$. It is shown that the Cayley structures are non-integrable, their basic geometric characteristics are calculate…

In this paper we study the geometry of metric spheres in the curve complex of a surface, with the goal of determining the "average" distance between points on a given sphere. Averaging is not technically possible because metric spheres in the curve complex are countably infinite and do not support any invariant probabi…

Reduces connectivity problem for genus-4 Heegaard surface in 3-sphere.

Characterizes simplicial complexes embedding into spheres with few vertices.

A filling Dehn sphere $Σ$ in a closed 3-manifold $M$ is a sphere transversely immersed in $M$ that defines a cell decomposition of $M$. Every closed 3-manifold has a filling Dehn sphere. The Montesinos complexity of a $3$-manifold $M$ is defined as the minimal number of triple points among all the filling Dehn spheres …

New insights into Khovanov homology complexity and topological structure.

Sphere-bases for simplicial and cubical complexes are constructed and analyzed.

The paper characterizes Whitney and contact Whitney spheres in complex and Sasakian space forms.

We introduce a notion of complexity for Sefiert homology spheres by establishing a correspondence between lattice point counting in tethrahedra and the Heegaard-Floer homology. This complexity turns out to be equivalent to a version of Casson invariant and it is monotone under a natural partial order in the set of Seif…

New 2D complex hyperbolic structures found on sphere orbibundles.

In this paper, we solve in the negative the following problem : Is there any complex structure on the sphere S^6?

In this paper, the singular-value decomposition theory of complex matrices is explored to study constantly curved 2-spheres minimal in both $\mathbb{C}P^n$ and the hyperquadric of $\mathbb{C}P^n$. The moduli space of all those noncongruent ones is introduced, which can be described by certain complex symmetric matrices…

We consider an embedding of a $2$-dimensional CW complex into the $3$-sphere, and construct it's dual graph. Then we obtain a homogeneous system of linear equations from the $2$-dimensional CW complex in the first homology group of the complement of the dual graph. By checking that the homogeneous system of linear equa…

Researchers prove the automorphism group of a sphere complex matches the mapping group of a graph.

We give a new proof that the sphere S^6 does not admit an integrable orthogonal complex structure, as in \cite{LeBrun}, following the methods from twistor theory. We present the twistor space of a pseudo-sphere S^{2n}_{2q}=SO_{2p+1,2q}/SO_{2p,2q} as a pseudo-Kähler symmetric space. We then consider orthogonal complex s…

Study shows non-existence of almost complex structures on certain sphere bundles over complex projective spaces.

We study the space of deformations of a smooth foliation of the 5-sphere by complex manifolds

New interpretation of complex hyperbolic form as Weil-Petersson form.

Handel and Mosher have proved that the free splitting complex FS for the free group is Gromov hyperbolic. This is a deep and much sought-after result, since it establishes FS as a good analogue of the curve complex for surfaces. We give a shorter alternative proof of this theorem, using surgery paths in Hatcher's spher…

In this paper, we prove some differentiable sphere theorems and topological sphere theorems for submanifolds in Kähler manifold, especially in complex space forms.

We explain an error in our paper "A smooth foliation of the 5-sphere by complex surfaces", Ann. Math 156 (2002), p.915-930.

The paper provides examples of keen weakly reducible bridge spheres for links in b-bridge position.

We study the existence or not of harmonic diffeomorphisms between certain domains in the Euclidean 2-sphere. In particular, we show harmonic diffeomorphisms from circular domains in the complex plane onto finitely punctured spheres, with at least two punctures. This result follows from a general existence theorem for m…

New theorem links tropical phased matroids to higher-dimensional spheres.

In this short note, we review the well-known result that there is no orthogonal complex structure on the 6-sphere with respect to the round metric.

This note constructs complex structures on specific isoparametric hypersurfaces.