




Geometric Topology / Topologie géometrique (Alex Chigogidze and Ed Tymchatyn, Organizers)
 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®R^{n} 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 nonseparating
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 fg about 0 is > = k1.
 BEVERLY BRECHNER, University of Florida, Florida, USA
Another approach to the HilbertSmith conjecture

The HilbertSmith (HS) conjecture states that a padic 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 lowdimensional fibers

Find the conditions under which a mapping is a locally trivial
fibrationthis 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 lowdimensional (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 ehomeomorphism).
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
4manifolds as PL fibrators

A closed PL nmanifold N is said to be a codimensionk
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 4manifolds N which are nontrivial connected sums
are codimension5 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
pi_{1}(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 326118105, 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 x_{0} Î 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. E. Michael proved that if E is a locally compact and
noncompact space then E×0 is a perfect retract of the product
E×[0,1) if and only if the onepoint 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 shapetheoretic terms. Michael stated the
following questions:
1. 1. Is a compact metric ARX strictly contractible to any x_{0} Î X?
1. 2. Is a compact metric ARX strictly contractible to any x_{0} Î X for which X\{x_{0}} 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 2polyhedra
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 h^{n} such
that "x Î X d(x,h^{n}(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
[`({h^{n}})] 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
[`({h^{n}})] 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
pathconnected and semilocally simply connected space an unramified map
is a covering space. We also provide a `generic' example of an unramified
map of locally pathconnected 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
pathconnected 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 CWcomplex. A space X is said to have extension
dimension £ [L] (notation: edimX £ [L]) if any mapping
of its closed subspace A Ì X into L admits an extension to the
whole space. We prove extensiondimensional 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 edimX £ [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 CWcomplex such that [L] £ [S^{n}] for some n. Let X be a paracompact space of extension dimension edimX £ [L]. Suppose that multivalued mapping F: X® Y into a complete metric space Y admits a lower [L]continuous, complete, and fiberwise [L]connected nfiltration F_{0} Ì F_{1} Ì ¼ Ì F_{n} Ì F. If f: A® Y is a continuous singlevalued selection of F_{0} over a closed subspace A Ì X, then there exists a continuous singlevalued selection g: X® Y of the mapping F such that g_{A}=f.
 KAZUHIRO KAWAMURA, University of Tsukuba, Institute of Mathematics, Tsukuba,
Ibaraki 3058571, Japan
Suspensions of the Hawaiian earring

The Hawaiian earring is a compact subset of the plane which is the
onepoint union of countably many simple closed curves. We consider
the unreduced nfold 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 ×(1x) 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
twoshift.
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 padic
action

Let G be a padic action on an nmanifold, 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 s_{2}(A) = A Ì T on which the map s_{2}_{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 s_{2} include (1) any rotational
s_{2}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 n1 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 semiconjugate, under m, to a (n1)to1 covering
map s_{n1}^{*} of a circle, and s_{n1}^{*} is topologically
conjugate to s_{n1}.
The Reduction Theorem allows us to relate rotational invariant
sets under s_{n} to those with the same rotation number under
s_{n1}. We will discuss s_{3} 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
Inchannels 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 nonseparating 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 inchannel.
 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 emappings 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 ndimensional surjective map of
metrizable spaces. It is shown that if Y is a Cspace, then:
(1) there exists a map g: X®I^{n} with dim(f×g)=0; (2) for every 0 £ k £ n1 there exists an
F_{s}subset A_{k} of X such that dimA_{k} £ k and the
restriction f(X\A_{k}) is nk1dimensional. 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 finitedimensional.
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 finitedimensional spaces

This is a joint work with V. Gutev. A characterization of
ndimensional 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 selfhomeomorphisms 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 selfhomeomorphism
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.

