Open Embedding Research Articles

Extending Maps in Hilbert Manifolds

The aim of the paper is to prove that if $M$ is a metrizable manifold modelled on a Hilbert space of dimension $\alpha \geq \aleph_0$ and $F$ is its $\sigma$-$Z$-set, then for every completely metrizable space $X$ of weight no greater than $\alpha$ and its closed subset $A$, for any map $f: X \to M$, each open cover $\mathcal{U}$ of $M$ and a sequnce $(A_n)_n$ of closed subsets of $X$ disjoint from $A$ there is a map $g: X \to M$ $\mathcal{U}$-homotopic to $f$ such that $g\bigr|_A = f\bigr|_A$, $g\bigr|_{A_n}$ is a closed embedding for each $n$ and $g(X \setminus A)$ is a $\sigma$-$Z$-set in $M$ disjoint from $F$. It is shown that if $f(\partial A)$ is contained in a locally closed $\sigma$-$Z$-set in $M$ or $f(X \setminus A) \cap \bar{f(\partial A)} = \empty$, the map $g$ may be taken so that $g\bigr|_{X \setminus A}$ be an embedding. If, in addition, $X \setminus A$ is a connected manifold modelled on the same Hilbert space as $M$ and $\bar{f(\partial A)}$ is a $Z$-set in $M$, then there is a $\mathcal{U}$-homotopic to $f$ map $h: X \to M$ such that $h\bigr|_A = f\bigr|_A$ and $h\bigr|_{X \setminus A}$ is an open embedding.

The period map for cubic fourfolds

The period map for cubic fourfolds takes values in a locally symmetric variety of orthogonal type of dimension 20. We determine the image of this period map (thus confirming a conjecture of Hassett) and give at the same time a new proof of the theorem of Voisin that asserts that this period map is an open embedding. An algebraic version of our main result is an identification of the algebra of SL (6,ℂ)-invariant polynomials on the representation space Sym 3(ℂ6)* with a certain algebra of meromorphic automorphic forms on a symmetric domain of orthogonal type of dimension 20. We also describe the stratification of the moduli space of semistable cubic fourfolds in terms of a Vinberg-Dynkin diagram.

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.

The period map for cubic threefolds

Allcock, Carlson and Toledo defined a period map for cubic threefolds which takes values in a ball quotient of dimension 10. A theorem of Voisin implies that this is an open embedding. We determine its image and show that on the algebraic level this amounts to identification of the algebra of with an explicitly described algebra of meromorphic automorphic forms on the complex 10-ball.

On the probabilistic domain invariance

Probabilistic version of the invariance of domain for contractive field and Schauder invertibility theorem are proved. As an application, the stability of probabilistic open embedding is established.

On complex analytic compactifications of complex parallelizable manifolds

We show that for a quotient of a complex Lie group by a lattice there does not exist any open equivariant holomorphic embedding into a larger complex space.

Characteristic cycles of constructible sheaves

In his paper [K], Kashiwara introduced the notion of characteristic cycle for complexes of constructible sheaves on manifolds: let X be a real analytic manifold, and F a complex of sheaves of C-vector spaces on X , whose cohomology is constructible with respect to a subanalytic strati cation; the characteristic cycle CC(F) is a subanalytic, Lagrangian cycle (with in nite support, and with values in the orientation sheaf of X ) in the cotangent bundle T ∗X . The de nition of CC(F) is Morse-theoretic. Heuristically, CC(F) encodes the in nitesimal change of the Euler characteristic of the stalks of F along the various directions in X . It tends to be di cult in practice to calculate CC(F) explicitly for all but the simplest complexes F; on the other hand, the characteristic cycle construction has good functorial properties. The behavior of CC(F) with respect to the operations of proper direct image, Verdier duality, and non-characteristic inverse image of F is well understood [KS]. In this paper, we describe the e ect of the operation of direct image by an open embedding. Combining our result with those that were previously known, we obtain descriptions of CC(Rf∗F) and CC(f∗F) – analogous to those in [KS] – for arbitrary morphisms f : X → Y in the semi-algebraic category, and complexes F with semi-algebraically constructible cohomology. In e ect, this provides an axiomatic characterization of the functor CC, at least in the semi-algebraic context. Our arguments do apply more generally in the subanalytic case, but because statements become quite convoluted, we shall not strive for the greatest degree of generality. As a concrete application, we consider the case of the ag manifold X of a complex semisimple Lie algebra g. Here the Weyl group W of g operates,

Menger manifolds homeomorphic to their 𝑛-homotopy kernels

We give a necessary and sufficient condition that an ( n + 1 ) (n + 1) -dimensional Menger manifold ( μ n + 1 {\mu ^{n + 1}} -manifold) is homeomorphic to its n-homotopy kernel. Such a μ n + 1 {\mu ^{n + 1}} -manifold is called a μ ∞ n + 1 \mu _\infty ^{n + 1} -manifold. We also prove the following results: (1) Each homeomorphism between two Z-sets in a μ ∞ n + 1 \mu _\infty ^{n + 1} -manifold M extends to an ambient homeomorphism of M onto itself if it is n-homotopic to id in M. (2) An n-homotopy equivalence between two μ ∞ n + 1 \mu _\infty ^{n + 1} -manifolds is n-homotopic to a homeomorphism. (3) Each map from a μ ∞ n + 1 \mu _\infty ^{n + 1} -manifold into a μ n + 1 {\mu ^{n + 1}} -manifold is n-homotopic to an open embedding.

On [formula omitted]∞ and Q∞-manifolds

We give a characterization of manifolds modeled on R ∞= dir lim or R nQ ∞=dir lim Q n , where Q is the Hilbert cube, and elementary short proofs of the Open Embedding Theorem for these manifolds and the following theorem generalizing the Stability Theorem: Each fine homotopy equivalence between these manifolds is a near homeomorphism. Moreover we establish the Open Embedding Approximation Theorem.

Each 𝑅^{∞}-manifold has a unique piecewise linear 𝑅^{∞}-structure

R. E. Heisey introduced piecewise linear R ∞ {{\mathbf {R}}^\infty } -structures and defined piecewise linear R ∞ {{\mathbf {R}}^\infty } -manifolds. In this paper we show that two piecewise linear R ∞ {{\mathbf {R}}^\infty } -manifolds are isomorphic if they have the same homotopy type. From the Open Embedding Theorem for (topological) R ∞ {{\mathbf {R}}^\infty } -manifolds and this result, we have the title.

Constructing Locally Flat Embeddings of Infinite Dimensional Manifolds without Tubular Neighborhoods

All spaces in this paper will be separable and metric. A closed embedding i: M → TV is said to be locally flat (of codimension n) if for each x0 ∈ M there is an open set U in M containing xo and an open embedding h: U X Rn → N for which h(x, 0) = i(x), for all x ∈ U.

Manifolds modelled on $R\sp{\infty }$ or bounded weak-*\ topologies

Let R = lim Rn, and let B*(b*) denote the conjugate, B*, of a separable, infinite-dimensional Banach space with its bounded weak-* topology. We investigate properties of paracompact, topological manifolds M, N modelled on F, where F is either R. or B*(b*). Included among our results are that locally trivial bundles and microbundles over M with fiber F are trivial; there is an open embedding M M X F; and if M and N have the same homotopy type, then M X F and N X F are homeomorphic. Also, if U is an open subset of B*(b*), then U X B*(b*) is homeomorphic to U. Thus, two open subsets of B*(b*) are homeomorphic if and only if they have the same homotopy type. Our theorems about B*(b*)-manifolds, B*(b*) as above, immediately yield analogous theorems about B(b)-manifolds, where B(b) is a separable, reflexive, infinite-dimensional Banach space with its bounded weak topology. 0. Introduction. Let R' = lim Rn, and let B*(b*) denote the conjugate, B*, of a separable, infinite-dimensional Banach space with its bounded weak-* topology. The bounded weak-* topology is the finest topology agreeing with the weak-* topology on bounded sets. (The weak-* topology on B* is the smallest topology on which all the linear functionals {xlIx E B} are continuous where A(a) = a(x).) We investigate properties of paracompact, topological manifolds, M, N, modelled on F, where F is either R' or B*(b*). Included among our results are that locally trivial bundles and microbundles over M with fiber F are trivial; there is an open embedding M M x F; and if M and N have the same homotopy type, then M x F is homeomorphic to N x F. We also show that if U is open in B*(b*), then U x B*(b*) is homeomorphic to U. Thus, two open subsets of B*(b*) are homeomorphic if and only if they have the same homotopy type. Any theorem about B*(b*)-manifolds, B*(b*) as above, yields an analogous Received by the editors May 10, 1973 and, in revised form, April 17, 1974. AMS (MOS) subject classifications (1970). Primary 58B05; Secondary 46A99.

Rings of quotients of topological rings

This paper is concerned primarily with the question of extending the topology of a topological ring to its complete ring of quotients so that the given ring is an open subring of the quotient ring. This type of problem was considered in a very special case by Warner [6]. General topological embeddings have been studied by End6 [1]. Both of these papers are concerned with the classical rather than the complete ring of quotients. More specifically, Warner requires that his rings have no zero-divisors. We have attempted to generalize these results by requiring only that our rings have an identity element and by working in the complete ring of quotients. In § 1 a description is given of a subring, C(R), of the complete ring of quotients which is maximum with respect to the property that the given ring may be embedded in it as an open subring. The following sections are devoted to properties of C(R) and its application to the study of the classical ring of quotients. In particular, the product theorem of Utumi [4; Proposition 9, p. 100] and the open embeddability criterion of Warner [6; Theorem 5, p. 59] are generalized. 1. The Maximum Open Embedding