Extension theory of separable metrizable spaces with applications to dimension theory

The paper deals with generalizing several theorems of the covering dimension theory to the extension theory of separable metrizable spaces. Here are some of the main results:

Generalized Eilenberg-Borsuk Theorem. Let be a countable CW complex. If is a separable metrizable space and is an absolute extensor of for some CW complex , then for any map , closed in , there is an extension of over an open set such that .

Theorem. Let be countable CW complexes. If is a separable metrizable space and is an absolute extensor of , then there is a subset of such that and .

Theorem. Suppose are countable, non-trivial, abelian groups and 0$">. For any separable metrizable space of finite dimension 0$">, there is a closed subset of with for .

Theorem. Suppose is a separable metrizable space of finite dimension and is a compactum of finite dimension. Then, for any , , there is a closed subset of such that and .

Theorem. Suppose is a metrizable space of finite dimension and is a compactum of finite dimension. If and are connected CW complexes, then

     

Cohomological dimension and approximate limits

Approximate (inverse) systems of compacta have been useful in the study of covering dimension, dim, and cohomological dimension over an abelian group , . Such systems are more general than (classical) inverse systems. They have limits and structurally have similar properties. In particular, the limit of an approximate system of compacta satisfies the important property of being an approximate resolution. We shall prove herein that if is an abelian group, a compactum is the limit of an approximate system of compacta , , and for each , then .

     

Cohomological approach to asymptotic dimension

We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension asdim _Z X of metric spaces. We show that it agrees with the asymptotic dimension asdim X when the later is finite. Then we use this fact to construct an example of a metric space X of bounded geometry with finite asymptotic dimension for which asdim(X × R) = asdim X. In particular, it follows for this example that the coarse asymptotic dimension defined by means of Roe’s coarse cohomology is strictly less than its asymptotic dimension.      

Extension dimension for paracompact spaces

We prove existence of extension dimension for paracompact spaces. Here is the main result of the paper:

Theorem. Suppose X is a paracompact space. There is a CW complex K such that

(a) K is an absolute extensor of X up to homotopy,

(b) If a CW complex L is an absolute extensor of X up to homotopy, then L is an absolute extensor of Y up to homotopy of any paracompact space Y such that K is an absolute extensor of Y up to homotopy.

The proof is based on the following simple result (see Theorem 1.2).

Theorem. Let X be a paracompact space. Suppose a space Y is the union of a family {Y_s}_sS of its subspaces with the following properties:

(a) Each Y_s is an absolute extensor of X,

(b) For any two elements s and t of S there is uS such that Y_sY_tY_u.

If f :A→Y is a map from a closed subset A to Y such that A=_sSInt_A(f⁻¹(Y_s)), then f extends over X.

That result implies a few well-known theorems of classical theory of retracts which makes it of interest in its own.     

Hereditarily aspherical compacta

The notion of (strongly) hereditarily aspherical compacta introduced by Daverman (1991) is modified. The main results are: Theorem. If is a hereditarily aspherical compactum, then ANR. In particular, is strongly hereditarily aspherical.

Theorem. Suppose is a cell-like map of compacta and is shape aspherical for each closed subset of . Then

1.

Y is hereditarily shape aspherical,

2.

is a hereditary shape equivalence,

3.

.

Theorem. Suppose is a group containing integers. Then the following conditions are equivalent:

1.

and ,

2.

.

Theorem. Suppose is a group containing integers. If and , then is hereditarily shape aspherical.

Theorem. Let be a two-dimensional, locally connected and semilocally simply connected compactum. Then, for any compactum

     

Compactifications and universal spaces in extension theory

We show that for each countable simplicial complex the following conditions are equivalent:

• iff for any space .
• There exists a -invertible map of a metrizable compactum with onto the Hilbert cube.

     

Products of Michael spaces and completely metrizable spaces

For disjoint subsets of the Michael space has the topology obtained by isolating the points in and letting the points in retain the neighborhoods inherited from . We study normality of the product of Michael spaces with complete metric spaces. There is a ZFC example of a Lindelöf Michael space , of minimal weight , with Lindelöf but with not normal. ( denotes the countable product of a discrete space of cardinality .) If denotes , the normality of implies the normality of for any complete metric space (of arbitrary weight). However, the statement `` normal implies normal' is axiom sensitive.

     

Small inductive dimension of completions of metric spaces. II

Extending the results of a previous paper under the same title we show that, under , .

     

Covering dimension and finite spaces

Finite topological spaces, that is spaces with a finite number of points, have a wide range of applications in many areas such as computer graphics and image analysis. In this paper we study the covering dimension of a finite topological space. In particular, we give an algorithm for computing the covering dimension of a finite topological space using matrix algebra.     

Sobolev spaces, dimension, and random series

We investigate dimension-increasing properties of maps in Sobolev spaces; we obtain sharp results with a random process somewhat like Brownian motion.

     

Algebra of dimension theory

The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory, this algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes of graded groups . There are two geometric interpretations of these equivalence classes: 1) For pointed CW complexes and , if and only if the infinite symmetric products and are of the same extension type (i.e., iff for all compact ). 2) For pointed compact spaces and , if and only if and are of the same dimension type (i.e., for all Abelian groups ).

Dranishnikov's version of the Hurewicz Theorem in extension theory becomes for all simply connected .

The concept of cohomological dimension of a pointed compact space with respect to a graded group is introduced. It turns out iff for all . If and are two positive graded groups, then if and only if for all compact .

     

Small inductive dimension of completions of metric spaces

We construct a 0-dimensional metric space which under a special set-theoretic assumption, denoted in the paper as S(), does not have a 0-dimensional completion. Shortly after the submission of the paper for publication R. Dougherty has shown the consistency of S(). (S() disagrees with the continuum hypothesis.)

     

Resolutions for metrizable compacta in extension theory

We prove a -resolution theorem for simply connected CW- complexes in extension theory in the class of metrizable compacta . This means that if is a connected CW-complex, is an abelian group, , , for , and (in the sense of extension theory, that is, is an absolute extensor for ), then there exists a metrizable compactum and a surjective map such that:

(a) is -acyclic,

(b) , and

(c) .

This implies the -resolution theorem for arbitrary abelian groups for cohomological dimension when . Thus, in case is an Eilenberg-MacLane complex of type , then (c) becomes .

If in addition , then (a) can be replaced by the stronger statement,

(aa) is -acyclic.

To say that a map is -acyclic means that for each , every map of the fiber to is nullhomotopic.

     

The image of the Thom map for Eilenberg-MacLane spaces

Fundamental classes in cohomology of Eilenberg-MacLane spaces are defined. The image of the Thom map from cohomology to mod- cohomology is determined for arbitrary Eilenberg-MacLane spaces. This image is a polynomial subalgebra generated by infinitely many elements obtained by applying a maximum number of Milnor primitives to the fundamental class in mod- cohomology. This subalgebra in mod cohomology is invariant under the action of the Steenrod algebra, and it is annihilated by all Milnor primitives. We also show that cohomology determines Morava cohomology for Eilenberg-MacLane spaces.

     

Mean dimension and AH-algebras with diagonal maps

Mean dimension for AH-algebras with diagonal maps is introduced. It is shown that if a simple unital AH-algebra with diagonal maps has mean dimension zero, then it has strict comparison on positive elements. In particular, the strict order on projections is determined by traces. Moreover, a lower bound of the mean dimension is given in term of Toms' comparison radius. Using classification results, if a simple unital AH-algebra with diagonal maps has mean dimension zero, it must be an AH-algebra without dimension growth. Two classes of AH-algebras with diagonal maps are shown to have mean dimension zero: the class of AH-algebras with at most countably many extremal traces, and the class of AH-algebras with numbers of extreme traces which induce same state on the _K0${\mathrm{K}}_0$-group being uniformly bounded (in particular, this class includes AH-algebras with real rank zero).     

Deformation theorems on non‐metrizable vector spaces and applications to critical point theory

Let E be a Banach space and Φ : E → ? a ??¹‐functional. Let ?? be a family of semi‐norms on E which separates points and generates a (possibly non‐metrizable) topology ??_?? on E weaker than the norm topology. This is a special case of a gage space, that is, a topological space where the topology is generated by a family of semi‐metrics. We develop some critical point theory for Φ : (E, ??) → ?. In particular, we prove deformation lemmas where the deformations are continuous with respect to ??_??. In applications this yields a gain in compactness when Φ does not satisfy the Palais–Smale condition because one can work with the weak topology. We also prove some foundational results on gage spaces. In particular, we introduce the concept of Lipschitz continuity in this setting and prove the existence of Lipschitz continuous partitions of unity. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hausdorff dimension and doubling measures on metric spaces

Volberg and Konyagin have proved that a compact metric space carries a nontrivial doubling measure if and only if it has finite uniform metric dimension. Their construction of doubling measures requires infinitely many adjustments. We give a simpler and more direct construction, and also prove that for any , the doubling measure may be chosen to have full measure on a set of Hausdorff dimension at most .