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Geometric Topology / Topologie géometrique
(Alex Chigogidze and Ed Tymchatyn, Organizers)

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SERGEI M. AGEEV, University of Saskatoon, Saskatoon, Saskatchewan  S7N 5E6
Geometric models of the Chigogidze and Dranishnikov resolutions

The mentioned above resolutions are constructed in explicit form as restriction of the linear map R®Rⁿ on appropriate subsets of R^m.

HAROLD BELL, University Of Cincinnati, Cincinnati, Ohio  45221, USA
On relating winding numbers and the fixed point index

The following conjecture and it's relationship to the non-separating plane fixed point problem will be discussed. Suppose A and B are disjoint arcs, p is a point that is neither in A or in B, k is a positive integer, f maps the unit interval onto A, and g maps the unit interval onto B. If both the winding number of f about p and the winding number of g about p are > = k then the winding number of f-g about 0 is > = k-1.

BEVERLY BRECHNER, University of Florida, Florida, USA
Another approach to the Hilbert-Smith conjecture

The Hilbert-Smith (H-S) conjecture states that a p-adic group cannot act effectively on a manifold. I will give an expository talk discussing some work of my students and myself toward a solution for n=3, as well as a possibly inductive approach for the general case. I will also discuss the open problems that need to be solved in order to complete such work.

NIKOLAY BRODSKIY, University of Saskatchewan, Saskatoon, Saskatchewan
Sections of maps with low-dimensional fibers

Find the conditions under which a mapping is a locally trivial fibration-this general problem is one of the interesting problems in topology. The obvious necessary condition is to have constant (up to a homeomorphism) fibers.

Shchepin's Conjecture. A Serre fibration with a metric locally arcwise connected base is locally trivial if it has a low-dimensional (of dimension n £ 4) compact manifold as a constant fiber.

We made a first step toward proving Shchepin's Conjecture in dimensions n=2 and n=3. A section of continuous mapping f: X®Y is continuous mapping s: Y® X such that f°s=id_Y. The mapping f : X ® Y is said to be topologically regular provided that if e > 0 and y Î Y, then there is a positive number d such that dist(y,y¢) < d, y¢ Î Y, implies that there is a homeomorphism of f^-1(y) onto f^-1(y¢) which moves no point as much as e (i.e. an e-homeomorphism).

Theorem. Let f : X ® Y be a topologically regular mapping of metric compacta with fibers homeomorphic to some compact manifold of dimension 2 or 3. If Y Î ANR, then any section of f over closed subset A Ì Y can be extended to a section of f over some neighborhood of A.

Also we provide some conditions under which the mapping f admits global section.

ROBERT J. DAVERMAN, University of Tennessee, Tennessee, USA
4-manifolds as PL fibrators

A closed PL n-manifold N is said to be a codimension-k orientable PL fibrator if every PL map p: M ® B from an orientable PL (n+k)-manifold M to a polyhedron B, where all fibers p^-1(B) are essentially copies of N, necessarily is an approximate fibration. Assembled in the belief that quick and easy detection of approximate fibrations is useful, the main result strives to establish that 4-manifolds N which are nontrivial connected sums are codimension-5 orientable PL fibrators. Almost invariably fundamental groups which are nontrivial free products posses algebraic features extremely useful for analysis here; when one of the summands of N is simply connected, the fibrator conclusion holds in case pi₁(N) is normally cohopfian and sparsely Abelian, the desirable algebraic features just mentioned, and also is residually finite. This represent work done jointly with Yongkuk Kim and Young Ho Im.

ALEXANDER N. DRANISHNIKOV, University of Florida, Gainesville, Florida  32611-8105, USA
On asymptotic dimension

The asymptotic dimension asdim was introduced by Gromov as a large scale analog of a local notion of the covering dimension. G. Yu proved the Novikov higher signature conjecture for groups G having asdimG < ¥. We prove asymptotic finite dimensionality of groups obtained from asymptotically finite dimensional groups by the standard constructions such as Serre's graph of groups. We discuss the Alexandroff's problem in the large scale geometry and a recent counterexample to it based on expanders.

JERZY DYDAK, University of Tennessee, Knoxville, Tennessee  37996, USA
Strict contractibility

A space X is strictly contractible to a point x₀ Î X if there exists a homotopy H:X×[0,1]® X starting at the identity such that H(x,t)=x₀ if and only if either x=x₀ or t=1. E. 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^*=EÈx^* of E is strictly contractible to x^*. We answer some questions posed by E. Michael in and we characterize strictly contractible ANRs in shape-theoretic terms. Michael stated the following questions:

1. 1. Is a compact metric ARX strictly contractible to any x₀ Î X?

1. 2. Is a compact metric ARX strictly contractible to any x₀ Î X for which X\{x₀} is AR?

We show that even in the case of compact polyhedra the answer to two questions of Michael is negative. However in the case of 2-polyhedra the answer to the second question is positive. Also, every collapsible polyhedron is strictly contractible to any point.

PAUL FABEL, Mississippi State University, Mississippi, Mississippi State  39762, USA
Characterizations of almost periodic homeomorphisms of metrizable spaces

Suppose h: X® X is a homeomorphism of the metric space (X,d).

If "e > 0 there exists an integer N so that each block of N consecutive iterates of h contains a map hⁿ such that "x Î X d(x,hⁿ(x)) < e then h is almost periodic.

If the orbit closure of each compactum B Ì X under the action of h is compact, and if for each compact invariant subset A Ì X, h_A: A® A is almost periodic then h is compactly almost periodic.

These are similar but distinct notions and we will offer various examples and characterizations.

Given a homeomorphism h of a metrizable space X, under what conditions is h compactly almost periodic? The answer is precisely if [`({hⁿ})] is a compact subspace of C(X,X) in the compact open topology.

Given a compactly almost periodic homeomorphism h of a metrizable space X, under what conditions does there exist a metric d on X such that h is almost periodic? The answer is precisely if [`({hⁿ})] is a metrizable subspace of C(X,X) in the compact open topology.

JONATHAN FUNK, Apartment 3, 282 Sherbrooke St. West, Montreal, Quebec  H2X 1X9
Unramified maps

We call a map of topological spaces unramified if it is both a sheaf space and a cosheaf space. We show that over a locally path-connected and semi-locally simply connected space an unramified map is a covering space. We also provide a `generic' example of an unramified map of locally path-connected spaces (which is onto and whose domain space is connected) that is not a covering space. We show that the category of unramified maps is a homotopy invariant of locally path-connected spaces.

(joint work with E. D. Tymchatyn)

ALEJANDRO ILLANES, Universidad Nacional Autnoma de México, Instituto de Matemáticas, Circuito Exterior, Cd. Universitaria, México, 04510, D.F.
Continua determined by their hyperspaces

Let X be a metric continuum. We consider the hyperspace, 2X, of all closed subsets of X, the hyperspace of subcontinua, C(X), of X and the hyperspace, Fn(X), of finite subsets of X having at most n elements. A continuum X is determined by its hyperspace C(X) provided that if Y is a continuum and C(X) is homeomorphic to C(Y), then X is homeomorphic to Y. Similar definitions are given for the hyperspaces 2X and Fn(X). In this talk we will discuss some of the known classes of continua which are determined by one of its hyperspaces and we will present some open problems on this subject.

ALEXANDRE KARASSEV, University of Saskatchewan, Saskatoon, Saskatchewan
Approximations and selections of multivalued mappings

Let L be a CW-complex. A space X is said to have extension dimension £ [L] (notation: e-dimX £ [L]) if any mapping of its closed subspace A Ì X into L admits an extension to the whole space. We prove extension-dimensional versions of finite dimensional selection and approximation theorems. As applications, we obtain several results on extension dimension. In particular, we prove a version of Hurewicz theorem for extension dimension.

A pair of spaces V Ì U is said to be [L]-connected if for every paracompact space X of extension dimension e-dimX £ [L] and for every closed subspace A Ì X any mapping of A into V can be extended to a mapping of X into U.

A multivalued mapping F: X® Y is called [L]-continuous at a point (x,y) Î G_F of its graph if for any neighborhood Oy of the point y Î Y, there exist a neighborhood O¢y of the point y and a neighborhood Ox of the point x Î X such that for all x¢ Î Ox, the pair F(x¢)ÇO¢y Ì F(x¢)ÇOy is [L]-connected. We call a multivalued mapping lower [L]-continuous if it is [L]-continuous at every point of its graph.

Now we can state our main result.

Filtered Selection Theorem. Let L be a locally finite countable CW-complex such that [L] £ [Sⁿ] for some n. Let X be a paracompact space of extension dimension e-dimX £ [L].
Suppose that multivalued mapping F: X® Y into a complete metric space Y admits a lower [L]-continuous, complete, and fiberwise [L]-connected n-filtration F₀ Ì F₁ Ì ¼ Ì F_n Ì F. If f: A® Y is a continuous single-valued selection of F₀ over a closed subspace A Ì X, then there exists a continuous single-valued selection g: X® Y of the mapping F such that g|_A=f.

KAZUHIRO KAWAMURA, University of Tsukuba, Institute of Mathematics, Tsukuba, Ibaraki  305-8571, Japan
Suspensions of the Hawaiian earring

The Hawaiian earring is a compact subset of the plane which is the one-point union of countably many simple closed curves. We consider the unreduced n-fold suspension of the Hawaiian earring and discuss an attempt to obtain the information on the (n+1)-dimensional homotopy group.

JAMES E. KEESLING, University of Florida, Gainesville, Florida, USA
Adding machines and the quadratic family of maps

Let f_m = m×x ×(1-x) be the quadratic family of maps on [0,1]. Let a be a sequence of primes. Let D_a be the adding machine associated with this sequence of primes. Let f_a be the adding machine map on D_a. This map is important in dynamical systems theory. The topological entropy of f_a is 0 and every orbit is dense in D_a.

In joint work with Louis Block and Jian Hua Xie, we locate all possible embeddings of D_a in [0,1] such that the associated adding machine map commutes with f_m in the quadratic family. This is done by first showing how to locate adding machines in the full two-shift.

If D_a is embedded in [0,1] in any way, we show how to compute the minimum topological entropy of any extension of f_a as a map of [0,1] into [0,1]. Given our method of making this computation, it is shown that there are embeddings of D_a in [0,1] such that the minimum entropy of an extension of f_a to all of [0,1] into itself can be arbitrarily large. The ability to compute the minimum topological entropy of an extension of f_a also gives insight into the relationship between adding machines and the quadratic family.

JAMES MAISSEN, University of Florida, Florida, USA
Constructing locally connected invariant sets of p-adic action

Let G be a p-adic action on an n-manifold, M. We construct an invariant, regular open set with Property S, such that both its boundary and its complement also have Property S.

JOHN C. MAYER, University of Alabama at Birmingham
Rotational invariant sets for covering maps of the circle

Consider the unit circle T=R/Z. The map s_n:T ® T is defined by s_n(x)=nx mod 1. For n=2, the invariant sets s₂(A) = A Ì T on which the map s₂|_A acts like rotation have been described topologically and combinatorially by Bullett and Sentenac. We call an invariant set s_n(A)=A on which s_n|_A acts like rotation rotational; a rotation number, which is the average speed with which s_n|_A rotates A, can then be associated with A.

Known facts about s₂ include (1)  any rotational s₂-invariant set must lie within a closed semicircle; (2)  for each rotation number r Î [0,1), there is a unique minimal invariant set A Ì T with that rotation number; (3)  A is a periodic orbit if r is rational, and is a Cantor set, otherwise, ``spanning'' a semicircle.

What differences arise when we study n > 2? A theorem that lays the groundwork for studying these differences is the following:

Theorem 1 For n ³ 2, a s_n-invariant set A Ì T is rotational iff T\A contains n-1 disjoint, open intervals (called ``holes''), each of length [1/(n)].

In some sense, we can take these holes out one at a time, as the following Reduction Theorem shows:

Theorem 2 Consider s_n:T®T, and for a Î T, let J=(a,a+[1/(n)] mod 1). Then X=T \È_k=0^¥s^-k(J) contains a unique maximal invariant Cantor set C. Moreover, if m:T ®T is the monotone map which collapses the closure of each complimentary interval of C to a point, then s_n is semi-conjugate, under m, to a (n-1)-to-1 covering map s_n-1^* of a circle, and s_n-1^* is topologically conjugate to s_n-1.

The Reduction Theorem allows us to relate rotational invariant sets under s_n to those with the same rotation number under s_n-1. We will discuss s₃ as a concrete example.

This is joint work with Alexander Blokh, James Malaugh, Daniel Parris, and Lex Oversteegen of UAB.

LEX G. OVERSTEEGEN, Alabama at Birmingham, Birmingham, Alabama  35294, USA
In-channels for negatively oriented maps of the plane

This is a report on joint work with Bell, Fokkink, Mayer and Tymchatyn. We will give a brief history of the plane fixed point problem including all recent results. We have shown that all confluent (i.e. compositions of open and monotone maps) of the plane are either positively or negatively oriented. Holomorphic maps are prototypes of positively oriented maps. Moreover, any positively oriented map has a fixed point in a non-separating invariant subcontinuum. This generalizes an old result for orientation preserving homeomorphisms of the plane. In order to generalize this result to negatively oriented maps of the plane, we will show that any minimal counter example to the plane fixed point problem for a negatively oriented map must have at least one dense generalized in-channel.

JANUSZ R. PRAJS, Department of Mathematics, Idaho State University, Pocatello, Idaho  83209, USA
Confluent mappings and arc property of Kelley

The arc property of Kelley is a property similar to the property of Kelley but stronger. The continua guaranteed by the definition are additionally required to be arcwise connected. The property naturally appears when absolute retract for various classes of hereditarily unicoherent continua are studied. Some theorems will be presented that show a strong relation between arc property of Kelley and confluent mappings. Among other things it is shown that a continuum that admits confluent e-mappings onto locally connected continua for every e > 0 has the arc property of Kelley.

(joint work with Janusz J. Charatonik and Wlodzimierz J. Charatonik)

H. MURAT TUNCALI, Nipissing University, North Bay, Ontario  P1B 8L7
On dimensionally restricted maps

Let f: X® Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1)  there exists a map g: X®Iⁿ with dim(f×g)=0; (2)  for every 0 £ k £ n-1 there exists an F_s-subset A_k of X such that dimA_k £ k and the restriction f|(X\A_k) is n-k-1-dimensional. These are extensions of theorems by Pasynkov and Torunczyk, respectively, obtained under the additional assumption that X and Y are compact (resp., separable) metric spaces with Y being finite-dimensional. A generalization of a result due to Dranishnikov and Uspenskij about extensional dimension is also established.

This is a joint work with Vesko Valov (Nipissing University)

VESKO VALOV, Nipissing University, North Bay, Ontario  P1B 8L7
Selections and finite-dimensional spaces

This is a joint work with V. Gutev. A characterization of n-dimensional paracompact spaces via continuous selections avoiding Z_n-sets is given. Applications of this theorem are also given.

YOUCHENG ZHOU, Zhejian University
A note on the self-homeomorphisms of solenoid and bucket handle

It is first shown that the torsion subgroup of the dyadic solenoid is dense, We prove by using this fact that for any self-homeomorphism of solenoid (or bucket handle) there exists a homeomorphism with dense set of periodic points which is homotopic to the previous one. And the later homeomorphism is also pointwise recurrent.