Compact maps and quasi-finite complexes

The simplest condition characterizing quasi-finite CW complexes K is the implication XτhK⇒β(X)τK for all paracompact spaces X. Here are the main results of the paper:

Theorem 0.1. If{Ks}_s∈Sis a family of pointed quasi-finite complexes, then their wedge_?s∈SKsis quasi-finite.     

Extraordinary dimension of maps

We consider the extraordinary dimension dimL introduced recently by Shchepin [E.V. Shchepin, Arithmetic of dimension theory, Russian Math. Surveys 53 (5) (1998) 975-1069]. If L is a CW-complex and X a metrizable space, then dimLX is the smallest number n such that ΣnL is an absolute extensor for X, where ΣnL is the nth suspension of L. We also write dimLf?n, where is a given map, provided dimLf⁻¹(y)?n for every y∈Y. The following result is established: Supposeis a perfect surjection between metrizable spaces, Y a C-space and L a countable CW-complex. Then conditions (1)-(3) below are equivalent:

(1)

dimLf?n;

(2)

There exists a dense andGδsubsetGofC(X,In)with the source limitation topology such thatdimL(f×g)=0for everyg∈G;

(3)

There exists a mapis such thatdimL(f×g)=0;If, in addition, X is compact, then each of the above three conditions is equivalent to the following one;

(4)

There exists anFσsetA⊂Xsuch thatdimLA?n−1and the restriction mapf|(X?A)is of dimensiondimf|(X?A)?0.

     

A combinatorial approach to coarse geometry

Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one of them can be transformed to the other by contiguous factorizations of bonding maps and by taking infinite subsequences. This embedding can be realized by either Rips complexes or analogs of Roe?s anti-?ech approximations of spaces.In this model coarse n-connectedness of K={K₁→K₂→?} means that for each k there is m>k such that the bonding map from Kk to Km induces trivial homomorphisms of all homotopy groups up to and including n.The asymptotic dimension being at most n means that for each k there is m>k such that the bonding map from Kk to Km factors (up to contiguity) through an n-dimensional complex.Property A of G. Yu is equivalent to the condition that for each k and for each ?>0 there is m>k such that the bonding map from |Kk| to |Km| has a contiguous approximation g:|Kk|→|Km| which sends simplices of |Kk| to sets of diameter at most ?.     

On two problems in extension theory

In this note we introduce the concept of a quasi-finite complex. Next, we show that for a given countable simplicial complex L the following conditions are equivalent:

•

L is quasi-finite.

•

There exists a [L]-invertible mapping of a metrizable compactum X with e-dimX?[L] onto the Hilbert cube.Finally, we construct an example of a quasi-finite complex L such that its extension type [L] does not contain a finitely dominated complex.

     

Extension dimension and quasi-finite CW-complexes

We extend the definition of quasi-finite complexes from countable complexes to arbitrary ones and provide a characterization of quasi-finite complexes in terms of L-invertible maps and dimensional properties of compactifications. Several results related to the class of quasi-finite complexes are established, such as completion of metrizable spaces, existence of universal spaces and a version of the factorization theorem. Furthermore, we define UV(L)-spaces in the realm of metrizable spaces and show that some properties of UV(n)-spaces and UV(n)-maps remain valid for UV(L)-spaces and UV(L)-maps, respectively.     

Covering dimension and finite-to-one maps

Hurewicz characterized the dimension of separable metrizable spaces by means of finite-to-one maps. We investigate whether this characterization also holds in the class of compact F-spaces of weight c. Our main result is that, assuming the Continuum Hypothesis, an n-dimensional compact F-space of weight c is the continuous image of a zero-dimensional compact Hausdorff space by an at most n²-to-1 map.     

Relative Nielsen theory for noncompact spaces and maps

In this note, we generalize the various existing local and relative Nielsen type numbers to the setting of maps of noncompact ANR-pairs. Then we introduce general classes of admissible maps for which these numbers are well-defined. An application of these relative Nielsen numbers to differential equations is also given.     

Refinable and monotone maps revisited

Generalizing results by J. Ford, J. W. Rogers, Jr. and H. Kato we prove that (1) a map f from a G-like continuum onto a graph G is refinable iff f is monotone; (2) a graph G is an arc or a simple closed curve iff every G-like continuum that contains no nonboundary indecomposable subcontinuum admits a monotone map onto G.We prove that if bonding maps in the inverse sequence of compact spaces are refinable then the projections of the inverse limit onto factor spaces are refinable. We use this fact to show that refinable maps do not preserve completely regular or totally regular continua.     

Coincidence Nielsen numbers for covering maps for smooth manifolds

Let be maps between closed smooth manifolds of the same dimension, and let and be finite regular covering maps. If the manifolds are nonorientable, using semi-index, we introduce two new Nielsen numbers. The first one is the Linear Nielsen number NL(f,g), which is a linear combination of the Nielsen numbers of the lifts of f and g. The second one is the Nonlinear Nielsen number NED(f,g). It is the number of certain essential classes whose inverse images by p are inessential Nielsen classes. In fact, N(f,g)=NL(f,g)+NED(f,g), where by abuse of notation, N(f,g) denotes the coincidence Nielsen number defined using semi-index.     

On continua with homotopically fixed boundary

The paper presents two subcontinua of Rn, one Peano-continuum, and one cellular continuum with trivial fundamental group. Both of them have the remarkable property that neither the entire spaces nor (roughly speaking) any part of them is homotopy equivalent to a lower-dimensional space. This extends work of the last three authors and of Karimov from the planar case to the higher-dimensional case, but it also contains in the cellular case the first example with all these properties in dimension two.     

A geometric interpretation of Ungar's addition and of gyration in the hyperbolic plane

We present a geometric interpretation of the operation a⊕b and the gyration on the unit-disc as defined by A.A. Ungar. Using this geometric interpretation we show that the two known generalizations to the n-dimensional unit ball are identical. The interpretation in the plane leads us to the notion of outer-median of a triangle and we discuss some possible properties of this median.     

On pairwise Lindelöf bitopological spaces

In this paper, we shall continue the study of bitopological separation axioms begun by Kelly and obtained some results. Furthermore, we introduce two concepts of pairwise Lindelöf bitopological spaces and the properties for them are established. We also show that a pairwise Lindelöf space is not hereditary property.     

On dimensional properties of the generalized Smirnov's spaces

We show that the transfinite inductive dimensions modulo PP-trind and P-trInd introduced in M.G. Charalambous (1997) [2] differ by simple spaces, where P is the absolutely additive Borel class A(α) or the absolutely multiplicative Borel class M(α), 0?α<ω₁.     

On productively Lindelöf spaces

The class of spaces such that their product with every Lindelöf space is Lindelöf is not well-understood. We prove a number of new results concerning such productively Lindelöf spaces with some extra property, mainly assuming the Continuum Hypothesis.     

There is no upper bound of small transfinite compactness degree in metrizable spaces

In [J.M. Aarts, T. Nishiura, Dimension and Extensions, North-Holland, Amsterdam, 1993], Aarts and Nishiura investigated several types of dimensions modulo a class P of spaces. These dimension functions have natural transfinite extensions. The small transfinite compactness degree trcmp is such transfinite dimension function extending the small compactness degree cmp. We shall prove that there is no upper bound for trcmp in the class of metrizable spaces, i.e. for each ordinal number α there exists a metrizable space Xα such that trcmpXα=α. We also give a characterization of the dimension dim of a separable (compact) metrizable space in terms of the function cmp of the product of this space with a nowhere locally compact zero-dimensional factor.     

On semilocally simply connected spaces

The purpose of this paper is: (i) to construct a space which is semilocally simply connected in the sense of Spanier even though its Spanier group is non-trivial; (ii) to propose a modification of the notion of a Spanier group so that via the modified Spanier group semilocal simple connectivity can be characterized; and (iii) to point out that with just a slightly modified definition of semilocal simple connectivity which is sometimes also used in literature, the classical Spanier group gives the correct characterization within the general class of path-connected topological spaces.While the condition “semilocally simply connected” plays a crucial role in classical covering theory, in generalized covering theory one needs to consider the condition “homotopically Hausdorff” instead. The paper also discusses which implications hold between all of the abovementioned conditions and, via the modified Spanier groups, it also unveils the weakest so far known algebraic characterization for the existence of generalized covering spaces as introduced by Fischer and Zastrow. For most of the implications, the paper also proves the non-reversibility by providing the corresponding examples. Some of them rely on spaces that are newly constructed in this paper.     

On fundamental groups of quotient spaces

In classical covering space theory, a covering map induces an injection of fundamental groups. This paper reveals a dual property for certain quotient maps having connected fibers, with applications to orbit spaces of vector fields and leaf spaces in general.