Open Embeddings Research Articles

Representation of relative sheaf cohomology

Abstract We study the cohomology theory of sheaf complexes for open embeddings of topological spaces and related subjects. The theory is situated in the intersection of the general Čech theory and the theory of derived categories. That is to say, on the one hand the cohomology is described as the relative cohomology of the sections of the sheaf complex, which appears naturally in the theory of Čech cohomology of sheaf complexes. On the other hand it is interpreted as the cohomology of a complex dual to the mapping cone of a certain morphism of complexes in the theory of derived categories. We prove a “relative de Rham-type theorem” from the above two viewpoints. It says that, in the case the complex is a soft or fine resolution of a certain sheaf, the cohomology is canonically isomorphic with the relative cohomology of the sheaf. Thus the former provides a handy way of representing the latter. Along the way we develop various theories and establishes canonical isomorphisms among the cohomologies that appear therein. The second viewpoint leads to a generalization of the theory to the case of cohomology of sheaf morphisms. Some special cases together with applications are also indicated. Particularly notable is the application of the Dolbeault complex case to the Sato hyperfunction theory and other problems in algebraic analysis.

Lefschetz Open Book Embeddings of 4-Manifolds

In this article, we define the notion of a generalized open book of a n-manifold over the k−sphere Sk , k < n. We discuss Lefschetz open book embeddings of Lefschetz open books of closed oriented 4-manifolds into the trivial open book over S2 of the 7−sphere S7 . If X is the double of a bounded achiral Lefschetz fibration over D2 , then X naturally admits a Lefschetz open book given by the bounded achiral Lefschetz fibration. We show that this natural Lefschetz open book of X admits a Lefschetz open book embedding into the trivial open book over S2 of the 7−sphere S7 .

Equivariant completions of affine spaces

We survey recent results on open embeddings of the affine space $\mathbb{C}^n$ into a complete algebraic variety $X$ such that the action of the vector group $\mathbb{G}_a^n$ on $\mathbb{C}^n$ by translations extends to an action of $\mathbb{G}_a^n$ on $X$. We begin with the Hassett-Tschinkel correspondence describing equivariant embeddings of $\mathbb{C}^n$ into projective spaces and present its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag varieties and their degenerations, complete toric varieties, and Fano varieties of certain types. Bibliography: 109 titles.

Equivariant completions of affine spaces

Работа содержит обзор недавних результатов об открытых вложениях аффинного пространства $\mathbb{C}^n$ в полные алгебраические многообразия $X$, для которых действие векторной группы $\mathbb{G}_a^n$ на $\mathbb{C}^n$ параллельными переносами продолжается до действия $\mathbb{G}_a^n$ на $X$. В первой части работы мы подробно изучаем соответствие Хассетта-Чинкеля, описывающее эквивариантные вложения $\mathbb{C}^n$ в проективные пространства, и приводим его обобщение на случай вложений в проективные гиперповерхности. Последующие разделы посвящены изучению вложений в многообразия флагов и в их вырождения, в полные торические многообразия и в многообразия Фано определeнных типов. Библиография: 109 названий.

Co-dimension two spinal open book embeddings of 3-manifolds

In this paper, we show that every spinal open book decomposition of a closed oriented [Formula: see text]-manifold [Formula: see text] spinal open book embeds into a certain spinal open book decomposition of [Formula: see text], the connected sum of [Formula: see text] copies of the twisted [Formula: see text]-bundle over [Formula: see text], where [Formula: see text] depends on the spinal open book decomposition of [Formula: see text]. We also discuss spinal open book embeddings of a huge class of spinal open books of closed oriented [Formula: see text]-manifolds into the trivial spinal open book of the [Formula: see text]-sphere [Formula: see text]. Finally, we show that given a closed oriented [Formula: see text]-manifold [Formula: see text], there exists a spinal open book for [Formula: see text] such that [Formula: see text] spinal open book embeds into the trivial spinal open book of [Formula: see text] In particular, this gives another proof of Hirsch’s theorem which states that every closed orientable [Formula: see text]-manifold embeds in [Formula: see text]

Kähler–Einstein Metrics on Smooth Fano Symmetric Varieties with Picard Number One

Symmetric varieties are normal equivarient open embeddings of symmetric homogeneous spaces, and they are interesting examples of spherical varieties. We prove that all smooth Fano symmetric varieties with Picard number one admit Kähler–Einstein metrics by using a combinatorial criterion for K-stability of Fano spherical varieties obtained by Delcroix. For this purpose, we present their algebraic moment polytopes and compute the barycenter of each moment polytope with respect to the Duistermaat–Heckman measure.

Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces

The fundamental groups and homotopy decompositions of algebraic topology have applications in systems involving symmetry breaking with topological excitations. The main aim of this paper is to analyze the properties of homotopy decompositions in quotient topological spaces depending on the connectedness of the space and the fundamental groups. This paper presents constructions and analysis of two varieties of homotopy decompositions depending on the variations in topological connectedness of decomposed subspaces. The proposed homotopy decomposition considers connected fundamental groups, where the homotopy equivalences are relaxed and the homeomorphisms between the fundamental groups are maintained. It is considered that one fundamental group is strictly homotopy equivalent to a set of 1-spheres on a plane and as a result it is homotopy rigid. The other fundamental group is topologically homeomorphic to the first one within the connected space and it is not homotopy rigid. The homotopy decompositions are analyzed in quotient topological spaces, where the base space and the quotient space are separable topological spaces. In specific cases, the decomposed quotient space symmetrically extends Sierpinski space with respect to origin. The connectedness of fundamental groups in the topological space is maintained by open curve embeddings without enforcing the conditions of homotopy classes on it. The extended decomposed quotient topological space preserves the trivial group structure of Sierpinski space.

Open embeddings and pseudoflat epimorphisms

We characterize open embeddings of Stein spaces and of C∞-manifolds in terms of certain flatness-type conditions on the respective homomorphisms of function algebras.

On the deformation rigidity of smooth projective symmetric varieties with Picard number one

Symmetric varieties are normal equivariant open embeddings of symmetric homogeneous spaces and they are interesting examples of spherical varieties. The principal goal of this article is to study the rigidity under Kähler deformations of smooth projective symmetric varieties with Picard number one.

PARABOLIC COMPACTIFICATION OF HOMOGENEOUS SPACES

AbstractIn this article, we study compactifications of homogeneous spaces coming from equivariant, open embeddings into a generalized flag manifold $G/P$. The key to this approach is that in each case $G/P$ is the homogeneous model for a parabolic geometry; the theory of such geometries provides a large supply of geometric tools and invariant differential operators that can be used for this study. A classical theorem of Wolf shows that any involutive automorphism of a semisimple Lie group $G$ with fixed point group $H$ gives rise to a large family of such compactifications of homogeneous spaces of $H$. Most examples of (classical) Riemannian symmetric spaces as well as many non-symmetric examples arise in this way. A specific feature of the approach is that any compactification of that type comes with the notion of ‘curved analog’ to which the tools we develop also apply. The model example of this is a general Poincaré–Einstein manifold forming the curved analog of the conformal compactification of hyperbolic space. In the first part of the article, we derive general tools for the analysis of such compactifications. In the second part, we analyze two families of examples in detail, which in particular contain compactifications of the symmetric spaces $\mathit{SL}(n,\mathbb{R})/\mathit{SO}(p,n-p)$ and $\mathit{SO}(n,\mathbb{C})/\mathit{SO}(n)$. We describe the decomposition of the compactification into orbits, show how orbit closures can be described as the zero sets of smooth solutions to certain invariant differential operators and prove a local slice theorem around each orbit in these examples.

Equivariant embedding theorems and topological index maps

The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a factorisation. These factorisations are models for the topological index map. Under some assumptions concerning the existence of equivariant vector bundles, any smooth map admits a normal factorisation, and two such factorisations are unique up to a certain notion of equivalence. To prove this, we generalise the Mostow Embedding Theorem to spaces equipped with proper groupoid actions. We also discuss orientations of normally non-singular maps with respect to a cohomology theory and show that oriented normally non-singular maps induce wrong-way maps on the chosen cohomology theory. For K-oriented normally non-singular maps, we also get a functor to Kasparov's equivariant KK-theory. We interpret this functor as a topological index map.

On period maps that are open embeddings

For certain complex projective manifolds (such as K3 surfaces and their higher dimensional analogues, the complex symplectic projective manifolds) the period map takes values in a locally symmetric variety of type IV. It is often an open embedding and in such cases it has been observed that the image is the complement of a locally symmetric divisor. We explain that phenomenon and get our hands on the complementary divisor in terms of geometric data.

Maximality Properties of Stable Partial Spreads and of Shear Planes

Shear planes are special stable planes which can be derived from stable partial spreads by a construction analogous to the construction of (punctured) dual translation planes from spreads. We give a sufficient condition for the maximality of shear planes in the category of stable planes and open embeddings. The condition is expressed in terms of the defining partial spread. Moreover, we give several examples of connected and disconnected shear planes in dimension 4, 8 and 16 which are maximal stable planes and have large automorphism groups. Most of these examples are direct limits of open subplanes of dual projective translation planes.

Critical exponents for square lattice trails with a fixed number of vertices of degree 4

We prove several previously conjectured results about the number of n-edge trails and n-edge embeddings of Eulerian graphs, each with a fixed number, k, of degree 4 vertices, in the lattice 2. In particular, under the assumption that the relevant critical exponents exist, we prove that the difference between the critical exponent for closed trails (Eulerian graph embeddings) and that for self-avoiding circuits (polygons) is exactly k, the number of degree 4 vertices. Similarly, we prove that the difference between the critical exponent for either open trails or open Eulerian graph embeddings and that for self-avoiding walks is also k. These results are proved by establishing upper and lower bounds for the number of n-edge embeddings of closed (open) Eulerian graphs with k vertices of degree 4 in terms of the number of n-edge self-avoiding polygons (walks). The lower bounds are proved using a Kesten pattern theorem argument and the upper bounds are established by developing (based on a detailed case analysis) a method for removing vertices of degree 4 from an embedding by altering at most a constant (independent of n) number of vertices and edges of the embedding. The work presented here extends and improves the arguments first given in the work of Zhao and Lookman (1993 J. Phys. A: Math. Gen. 26 1067–76).

The abstract boundary—a new approach to singularities of manifolds

A new scheme is proposed for dealing with the problem of singularities in General Relativity. The proposal is, however, much more general than this. It can be used to deal with manifolds of any dimension which are endowed with nothing more than an affine connection, and requires a family C of curves satisfying a bounded parameter property to be specified at the outset. All affinely parametrised geodesics are usually included in this family, but different choices of family C will in general lead to different singularity structures. Our key notion is the abstract boundary or a-boundary of a manifold, which is defined for any manifold M and is independent of both the affine connection and the chosen family C of curves. The a-boundary is made up of equivalence classes of boundary points of M in all possible open embeddings. It is shown that for a pseudo-Riemannian manifold ( M , g) with a specified family C of curves, the abstract boundary points can then be split up into four main categories—regular, points at infinity, unapproachable points and singularities. Precise definitions are also provided for the notions of a removable singularity and a directional singularity. The pseudo-Riemannian manifold will be said to be singularity-free if its abstract boundary contains no singularities. The scheme passes a number of tests required of any theory of singularities. For instance, it is shown that all compact manifolds are singularity-free, irrespective of the metric and chosen family C . All geodesically complete pseudo-Riemannian manifolds are also singularity-free if the family C simply consists of all affinely parametrised geodesics. Furthermore, if any closed region is excised from a singularity-free manifold then the resulting manifold is still singularity-free. Numerous examples are given throughout the text. Problematic cases posed by Geroch and Misner are discussed in the context of the a-boundary and are shown to be readily accommodated.

Local Lie semigroups and open embeddings into global topological semigroups

The semigroup version of Lie's Third Fundamental Theorem asserts that each strictly positive cone and each finite-dimensional Lie wedge is a tangent wedge of some local semigroup. This paper investigates the question whether this local semigroup can be embedded into a global topological semigroup in such a fashion that the embedding preserves all existing products and its image is open. In the case of a strictly positive cone (in particular a finite-dimensional pointed cone) this question will be answered affirmatively. This result contrasts the situation of open embeddings into global subsemigroups of Lie groups, where counterexamples even in low-dimensional Lie groups occur. In the finite-dimensional case a homotopy-like congruence on the semigroup of conal curves induces a quotient semigroup (called conal path semigroup) which also solves the topological embedding problem.

Deformation of Open Embeddings of Q-Manifolds

Deformation of Open Embeddings of Q-Manifolds

Stability, open embeddings, and classification of bounded weak- ∗ manifolds

It is known that the following spaces are homeomorphic: (1) a separable, reflexive, infinite-dimensional Banach space with its bounded weak topology, (2) the conjugate of a separable, infinite-dimensional Banach space with its bounded weak- ∗ topology, and (3) Q ∞ = dir lim Q n , where Q is the illbert cube. Let F denote one of the above three spaces, and let M and N denote paraimpact, connected F manifolds. Here we prove that M × F is homeomorphic to M. Combined with previous work of the author this result implies that M, embeds as an open subset of F and that if M and N have the same homotopy type then they are homeomorphic.

Deformation of open embeddings of 𝑄-manifolds

We prove here for Hilbert cube manifolds the full analogue of the Černavski-Edwards-Kirby Theorem concerning the deformation Principle for open embeddings of topological manifolds.

On the embedding of compact domains

w 1. Introduction In an earlier paper [4], the author gave necessary and sufficient conditions for a topological left Ore domain to be topologically embeddable in its field of quotients. In w 2, we use the results of [4] to give a new proof of Warner's theorem on open embeddings [7]. In w 3, we prove that a commutative domain A which is a Q-ring with continuous quasi-inversion has a finer non-discrete ring topology in which the multiplicative semigroup A* = A- {0} is embeddable in its quotient group. We also prove that an infinite compact Hausdorff commutative domain has a finer non-discrete ring topology in which it is openly embeddable in its quotient field. These theorems generalize the following theorem of Rothman [5]. Theorem. A compact Hausdorff commutative domain A with identity has a finer non-discrete topology on A* for which A* is embeddable in its group of quotients. For completeness, we state the following result due to Tamari [6]. Theorem T. A commutative cancellative topological semigroup S is embeddable in its group of quotients if and only if its topology is invariantly uniformisable.