One-point Compactification Research Articles

Generalized Geometric Loop Groups of Complex Manifolds, Gaussian Quasi-Invariant Measures on Them and Their Representations

Loop groups are very important in differential geometry, algebraic topology, and theoretical physics [7, 9, 16, 37, 47], but nothing was known about Gaussian quasi-invariant differentiable measures on them. Only the simplest possible representations associated with path integrals were constructed for loop groups of the circle, i.e., for the manifold M = S1 and (real) Riemannian manifolds N [37]. On the other hand, quasi-invariant measures can be used for construction of regular unitary representations [30–32,35]. Moreover, quasi-invariant measures are helpful for investigation of a group itself. In the previous papers of the author [33, 34], loop groups of Riemannian manifolds M and N were investigated, where either M = S is an n-dimensional real sphere, n = 1, 2, . . . , or M = S∞ is the unit sphere in a real separable Hilbert space l2(R). This was progress in comparison with previous works of others authors, which considered only loop groups for the simplest case M = S1. This paper treats arbitrary complex separable connected metrizable manifolds M and N . For example, products of odd-dimensional real spheres S2n−1×S2m−1 can be endonved with structures of complex manifolds in different ways [29]. Other numerous examples of complex manifolds can be found in [25] and references therein; they are domains in Cn, the complex torus Cn/D, where D is a discrete additive subgroup of Cn generated by a basis τ1, . . . , τ2n of C n over R; the quotient space G/D of a complex Lie group G by a discrete subgroup D, submanifolds of the complex Grassmann manifold Gp,q(C), and also their different products and submanifolds. In general, there are complex compact manifolds which are not Kahler manifolds [28,36]. For construction of loop groups here, we use manifolds M with some mild additional conditions. When M is finite-dimensional over C, we assume that it is compact. This condition is not very restrictive, since each locally compact space admits the Alexandrov (one-point) compactification (see [13, Theorem 3.5.11]). When M is infinite-dimensional over C, it is assumed that M is embedded as a closed bounded subset in the corresponding Banach space XM over C. This is necessary for defining a group structure on the quotient space of a free loop space. The free loop space is considered as consisting of continuous functions f : M → N which are holomorphic on M \ M ′ and preserving distinguished points f(s0) = y0, where M ′ is a closed real submanifold depending on f of codimension codimRM ′ = 1, s0 ∈ M , and y0 ∈ N are distinguished points. There are two reasons to consider such a class of mappings. The first is the need to correctly define compositions of elements in the loop group (see below). The second is the isoperimetric inequality for holomorphic loops, which can impose the condition that the loop be constant on a sufficiently small neighborhood of s0 in M if this loop is in some small neighborhood of w0, where w0(M) := {y0} is a constant loop (see [23, Remark 3.2]). In this paper, loop groups of different classes are considered. Classes analogous to Gevrey classes of f : M \M ′ → N are considered for construction of dense loop subgroups and quasi-invariant measures. Henceforth, we consider only orientable manifolds M and N , since for a nonorientable manifold, there always exists its orientable double covering manifold (see [1, 6.5]). Loop commutative monoids with cancellation property are quotients of families of mappings f from M into a manifold N with f(s0) = y0

Examples concerning extensions of continuous functions

Given a space Y, let us say that a space X is a total extender for Y provided that every continuous map f :A→Y defined on a subspace A of X admits a continuous extension f ̃ :X→Y over X. The first author and Alberto Marcone proved that a space X is hereditarily extremally disconnected and hereditarily normal if and only if it is a total extender for every compact metrizable space Y, and asked whether the same result holds without any assumption of metrizability on Y. We demonstrate that a hereditarily extremally disconnected, hereditarily normal, non-collectionwise Hausdorff space X constructed by Kenneth Kunen is not a total extender for K, the one-point compactification of the discrete space of size ω 1. Under the assumption 2 ω 0 =2 ω 1 , we provide an example of a separable, hereditarily extremally disconnected, hereditarily normal space X that is not a total extender for K. Furthermore, using forcing we prove that, in the generic extension of a model of ZFC+MA( ω 1), every first-countable separable space X of size ω 1 has a finer topology τ on X such that ( X, τ) is still separable and fails to be a total extender for K. We also show that a hereditarily extremally disconnected, hereditarily separable space X satisfying some stronger form of hereditary normality (so-called structural normality) is a total extender for every compact Hausdorff space, and we give a non-trivial example of such an X.

P-compact groups as framed manifolds

We describe a natural way to associate to any p-compact group an element of the p-local stable stems, which, applied to the p-completion of a compact Lie group G, coincides with the element represented by the manifold G with its left-invariant framing. To this end, we construct a d-dimensional sphere S G with a stable G-action for every d-dimensional p-compact group G, which generalizes the one-point compactification of the Lie algebra of a Lie group. The homotopy class represented by G is then constructed by means of a transfer map between the Thom spaces of spherical fibrations over BG associated with S G .

One-point compactifications of ditopological texture spaces

In this paper a one-point compactification of a ditopological texture space is defined and a concept of local compactness introduced and shown to be additive in the class of ditopological texture spaces. An application to Hutton spaces is given. Here a compact extension of an Hutton space is obtained and a notion of local compactness is defined and shown to be productive in the class of L -topological spaces, where L is a Hutton algebra.

Topology of spaces of hyperbolic polynomials and combinatorics of resonances

In this paper we study the topology of the strata, indexed by number partitions λ, in the natural stratification of the space of monic hyperbolic polynomials of degreen. We prove stabilization theorems for removing an independent block or an independent relation in λ. We also prove contractibility of the ‘one-point compactifications of the strata indexed by a large class of number partitions, including λ=(k m, 1r), form≥2. Furthermore, we study the maps between the homology groups of the strata, induced by imposing additional relations (resonances) on the number partition λ, or by merging some of the blocks of λ.

Locally-generic Boolean algebras and cardinal sequences

Given a sequence of cardinals \( \theta \) of length less than \( \scr w_2 \), with each cardinal in the sequence being either \( \scr w \) or \( \scr w_1 \), we construct a \( \theta \)-poset (see Defnition 1 below) which, with a natural topology, becomes a locally-compact, Hausdorff, scattered space with cardinal sequence \( \theta \). The algebra of the clopen subsets of its one-point compactification yields, in turn, a superatomic Boolean algebra with \( \theta \) as its cardinal sequence. The posets are locallygeneric, that is, they are constructed generically over countable sets. This gives them additional chain properties, specially under \( \diamondsuit \) Under Martin's Axiom, the construction allows any cardinals \( \leq2^w \) in the sequence, provided it has length \( \leq{w_1} \) Finally, we modify a forcing argument of Baumgartner-Shelah [B-S], to build \( \theta \)-posets for any given cardinal sequence \( \theta \) of length \( w_2 \) with each cardinal in the sequence being either \( w \) or \( w_1 \).

On power homogeneous spaces

The notion of a Moscow space [A.V. Arhangel'skii, Comment. Math. Univ. Carolinae 24 (1983) 105–120] and a new notion of a weakly Klebanov space, introduced below, are applied to study topological properties of spaces X such that some power of X is homogeneous. Under this approach, a crucial role is played by a theorem of Y. Yajima [J. Math. Soc. Japan 36 (1984) 689–699]. Several general theorems are established that expose the general rules behind certain earlier more concrete results of E. van Douwen [Proc. Amer. Math. Soc. 69 (1978) 183–192] and V.I. Malychin [Proceedings of Leningrad International Topology Conference, Nauka, Leningrad, 1983, pp. 50–61]. In particular, it is proved that every Corson compact space X such that X τ is homogeneous, for some cardinal number τ, is first countable (Corollary 4.9). This shows why every power of the one-point compactification of an uncountable discrete space is not homogeneous [Proc. Amer. Math. Soc. 69 (1978) 183–192]. V.I. Malychin proved that ( ω 1+1) τ is not homogeneous for any τ [Proceedings of Leningrad International Topology Conference, Nauka, Leningrad, 1983, pp. 50–61]. This is covered by Corollary 4.12 below: if some power of a compact scattered space X is homogeneous, then X is countable. It is also shown that, on many occasions, βX is the only compactification of X that can be homogeneous or power homogeneous.

Compact semilattices with open principal filters

AbstractA locally compact semilattice with open principal filters is a zero-dimensional scattered space. Cardinal invariants of locally compact and compact semilattices with open principal filters are investigated. Structure of topological semilattices on the one-point Alexandroff compactification of an uncountable discrete space and linearly ordered compact semilattices with open principal filters are researched.

Rational homology of spaces of complex monic polynomials with multiple roots

This paper treats rational homology groups of one-point compactifications of spaces of complex monic polynomials with multiple roots. These spaces are indexed by number partitions. A standard reformulation in terms of quotients of orbit arrangements reduces the problem to studying certain triangulated spaces Xλ,μ.

A comparison of Vassiliev and Ziegler–Živaljević models for homotopy types of subspace arrangements

In this paper we represent the Vassiliev model for the homotopy type of the one-point compactification of subspace arrangements as a homotopy colimit of an appropriate diagram over the nerve complex of the intersection semilattice of the arrangement. Furthermore, using a generalization of simplicial collapses to diagrams of topological spaces over simplicial complexes, we construct an explicit deformation retraction from the Vassiliev model to the Ziegler–Živaljević model.

A characterization of locally compact spaces with homeomorphic one-point compactifications

Let X and Y be locally compact noncompact spaces. For a large class of closed linear subspaces, A and B, of C 0(X, R) and C 0(Y, R) , respectively, we show that there exist basically two types of diameter-preserving linear bijections Ψ :A→B : those (type 1) which can be written as Ψ( f)( y)= τ( f∘ ϕ)( y), where τ∈{−1,1} and ϕ :Y→X is a surjective homeomorphism, and those (type 2) which can be represented as Ψ(f)(y)= τ(f∘ϕ)(y)−τf(x 0), y≠y 0, −τf(x 0), y=y 0 for some x 0∈ X and y 0∈ Y, where ϕ :Y⧹{y 0}→X⧹{x 0} is a homeomorphism which can be extended to the one-point compactifications of Y and X. All these facts are proved as a consequence of a full description of the extreme points of the closed unit ball of the dual of C 0( X) and such subspaces endowed all with the diameter norm. Finally, we characterize the locally compact spaces with homeomorphic one-point compactifications as those which admit diameter-preserving linear bijections like the ones described above.

On questions of strict contractibility

A space X is strictly contractible to a point x 0∈ X if there exists a homotopy H :X×[0,1]→X starting at the identity such that H( x, t)= x 0 if and only if either x= x 0 or t=1. Michael proved that if E is a locally compact and non-compact space then E×0 is a perfect retract of the product E×[0,1) if and only if the one-point compactification E ∗=E∪x ∗ of E is strictly contractible to x ∗ . We answer some questions posed by Michael [Closed retracts and perfect retracts, Topology Appl., to appear] and we characterize strictly contractible ANRs in shape-theoretic terms.

Hausdorff compactifications of topological function spaces via the theory of continuous lattices

It is known from the theory of continuous lattices that if X is a locally compact Hausdorff space then the set LSC(X) of lowersemicontinuous functions defined on X with values on the extended real lineadmits a unique compact Hausdorff topology making the functional (f, g)to min(f, g) continuous, namely the Lawson topology of the continuouslattice LSC(X). It is natural to wonder whether the relative topologyon the subset C(X) of continuous functions is the compact-opentopology. Unfortunately, it turns out to be strictly weaker. But a relatedconstruction does produce a Hausdorff compactification of C(X). Weshow that if X is a locally compact Hausdorff space and Y is aHausdorff topological space which is perfectly embedded into a continuouslattice L endowed with the Scott topology, then the Lawson topologyon the continuous lattice LSC(X,L) of Scott continuous maps fromX to L induces the compact-open topology on the spaceC(X,Y) of continuous maps from X to Y. Thus, by takingthe closure of the image of C(X,Y) in LSC(X,L), one gets aHausdorff compactification of C(X,Y). Three particular cases are ofinterest. (1) If Y is the Euclidean real line one can take L as the lattice of compact connected subsets of the two-pointcompactification of Y ordered by reverse inclusion. In this case,C(X,Y) is already dense in LSC(X,L). (2) If Y is alocally compact Hausdorff space, one can take L as the compactsubsets of the one-point compactification of Y. (3) As a furtherparticular case of (2), if X and Y are compact Hausdorff, oneconcludes that the Vietoris topology on the closed subsets of the cartesianproduct of X and Y induces the compact-open topology onC(X,Y), by identifying continuous functions with their closed graphs, using the fact that the Lawson topology coincides with the Vietoristopology.

A tensor-hom adjunction in a topos related to vector topologies and bornologies

In this paper, N = N∪{∞} is the one-point compactification of the discrete space of natural numbers, M is the monoid of continuous maps f: N→ N such that f(∞)=∞, and M is the topos of M-sets. We define two sheaf subtoposes C and B of M and construct a tensor-hom adjunction between certain categories of modules in C and B . Finally, we prove that this construction induces an adjunction between adequate categories of topological and bornological real vector spaces.

Maximal norms on Banach spaces of continuous functions. A Corrigendum to Kalton and Wood's ‘Orthonormal systems in Banach spaces and their applications’

The purpose of this short note is to reformulate theorem 9·3 in [5] which is not correct as stated. We note that all other results in [5] are independent of that statement.The notation is the same as [5] with the sole exception that C0(S) will always denote the space of all real-valued continuous functions on the locally compact space S vanishing at infinity. As usual, αS stands for the one-point compactification of S. Recall that a norm ‖ · ‖ on a Banach space X is said to be maximal if there is no equivalent norm on X whose isometry group contains properly that of ‖ · ‖.

A Secondary Chern-Euler Class

Let xi be a smooth oriented vector bundle, with n-dimensional fibre, over a smooth manifold M. Denote by xi-hat the fibrewise one-point compactification of xi. The main purpose of this paper is to define geometrically a canonical element Upsilon(xi) in H^n(xi-hat,Q) (H^n(xi-hat,Z) tensor 1/2, to be more precise). The element \Upsilon(\xi) is a secondary characteristic class to the Euler class in the fashion of Chern-Simons.

Extensions of topological spaces with strongly-discrete remainder

The construction of the Alexandroff one-point compactification is extended to provide paracompact extensions of locally compact Hausdorff spaces with strongly-discrete remainder.

Holomorphic functions on symmetric Banach manifolds of compact type are constant

1. Throughout, we use C to denote the one-point compactification C∪{∞} of C. Let (Z, ‖ · ‖) denote a Banach space over the complex numbers C. Let L(Z) and L(Z) denote, respectively, the space of continuous linear and anti-linear (or conjugate linear) mappings from Z to Z endowed with the operator norm. We consider a continuous triple product {·, ·, ·} : Z3 → Z, which is symmetric and linear in the outer variables and conjugate linear in the middle variable. In particular, this implies

Transitivity of M-spaces and Wood's conjecture

For a wide class of categories of Banach spaces, we show that the existence of an (almost) transitive element implies the existence of a separable almost transitive element. We give some applications to C 0 ( L ) spaces and abstract M -spaces. Next, we prove that Wood's conjecture on almost transitivity of the norm in C 0 ( L ) can be reduced to the case in which the one-point compactification of L is metrizable. We construct a simple example of transitive M -space and we show the existence of almost transitive separable M -spaces that are isomorphic to C [0, 1]. These M -spaces have a rich M -structure and they are counterexamples for several questions about centralizers.

Singular compactifications of product spaces

Assume that both $X$ and $Y$ are non-compact locally compact spaces. Let $\delta(X\times Y)$ be a compactification of $X\times Y$ such that $\delta(X\times Y)\geq\omega X\times\omega Y$, where $\omega X$ and $\omega Y$ are the one-point compactifications of $X$ and $Y$, respectively. Then J. L. Blasco [2] proved the theorem that $\delta(X\times Y)$ is not a weakly singular compactification of $X\times Y$ if $X$ is pseudcompact. In this paper we give an alternative, simpler proof for the above theorem. Furthermore, in the case $X$ is either a non-separable metrizable space or a separable metrizable space with a non-compact quasi-component space $Q(X)$ and $d(Y)\leq d(X)$, where $d(X)$ is the density of $X$, for any compact space $S$ we establish a theorem that $X\times Y$ has a singular compactification with $S$ as a remainder if and only if $X$ has a singular compactification with $S$ as a remainder.