Related papers: A characterization of normal 3-pseudomanifolds wit…
Every indefinite binary form occurs as the Picard lattice of some K3-surface. The group of its isometries, or automorphs, coincides with the automorphism group of the K3-surface, but only up to finite groups. The classical theory of…
In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics: The pull-back metrics of wave fronts which admit only…
When two smooth manifold bundles over the same base are fiberwise tangentially homeomorphic, the difference is measured by a homology class in the total space of the bundle. We call this the relative smooth structure class. Rationally and…
The problem of resolution of singularities in positive characteristic can be reformulated as follows: Fix a hypersurface $X$, embedded in a smooth scheme, with points of multiplicity at most $n$. Let an $n$-sequence of transformations of…
We exhibit many examples of closed symplectic manifolds on which there is an autonomous Hamiltonian whose associated flow has no nonconstant periodic orbits (the only previous explicit example in the literature was the torus T^2n (n\geq 2)…
We discuss the complex geometry of two complex five-dimensional K\"ahler manifolds which are homogeneous under the exceptional Lie group $G_2$. For one of these manifolds rigidity of the complex structure among all K\"ahlerian complex…
The algebra of exterior differential forms on a regular 3-Sasakian 7-manifold is investigated, with special reference to nearly-parallel $G_2$ 3-forms. This is applied to the study of 3-forms invariant under cohomogeneity-one actions by…
We provide a complete geometric solution to the problem of differentiating simplicial manifolds, extending classical Lie theory and complementing existing homotopical and formal approaches within a unifying framework. First, we establish a…
The aim of this paper is to provide a discussion on current directions of research involving typical singularities of 3D nonsmooth vector fields. A brief survey of known results is presented. The main purpose of this work is to describe the…
We show that the number of double points of smoothly immersed 2-spheres representing certain homology classes of an oriented, smooth, closed, simply-connected 4-manifold X must increase with the complexity of corresponding h-cobordisms from…
Our first main result is a uniform bound, in every dimension $k \in \mathbb N$, on the topological Tur\'an numbers of $k$-dimensional simplicial complexes: for each $k \in \mathbb N$, there is a $\lambda_k \ge k^{-2k^2}$ such that for any…
The main invariant to study the combinatorics of a simplicial complex $K$ is the associated face ring or Stanley-Reisner algebra. Reisner respectively Stanley explained in which sense Cohen-Macaulay and Gorenstein properties of the face…
The goal of this book is to characterize algebraically the closed 4-manifolds that fibre nontrivially or admit geometries in the sense of Thurston, or which are obtained by surgery on 2-knots, and to provide a reference for the topology of…
In this paper, we provide a simple proof for the fact that two simplicial complexes are isomorphic if and only if their associated Stanley-Reisner rings, or their associated facet rings are isomorphic as $K$-algebras. As a consequence, we…
We show that every small resolution of a three-dimensional terminal hypersurface singularity can occur on a non-embeddable 1-convex manifold. We give an explicit example of a non-embeddable manifold containing an irreducible exceptional…
A special spine of a three-manifold is said to be poor if it does not contain proper simple subpolyhedra. Using the Turaev-Viro invariants, we establish that every compact three-dimensional manifold M with connected nonempty boundary has a…
We study the topology of closed, simply-connected, 6-dimensional Riemannian manifolds of positive sectional curvature which admit isometric actions by $SU(2)$ or $SO(3)$. We show that their Euler characteristic agrees with that of the known…
We prove a conjecture of Maulik, Pandharipande, and Thomas expressing the Gromov--Witten invariants of K3 surfaces for divisibility two curve classes in all genus in terms of weakly holomorphic quasimodular forms of level two. Then, we…
CR singularities of real 4-submanifolds in complex 3-space are classified by using local holomorphic coordinate changes to transform the quadratic coefficients of the real analytic defining equation into a normal form. The quadratic…
For a genus g handlebody H a simplicial complex, with vertices being isotopy classes of certain incompressible surfaces in H, is constructed and several properties are established. In particular, this complex naturally contains, as a…