If a homeomorphism on the bucket handle has an invariant composant, it has a fixed point in that composant. It follows that a homeomorphism on the bucket handle has at least two fixed points. This answers a question by Mahavier (Problem 120 of the Houston Problem Book). Our methods apply to general Knaster continua.

Screenability (the property that every open cover has a sigma- disjoint open refinement) was introduced and studied in R.H.Bing's 1951 paper on metrization. An interesting question from that era, first published in a 1955 paper of K. Nagami, asks whether normality and screenability together are equivalent to paracompactness. The problem was restated as Classic Problem III in the problem section of Topology Proceedings in 1976. In 1983, M.E.Rudin constructed a normal, screenable, nonparacompact space from V=L. We are going to present the following result.

Theorem 1. There is a normal, screenable, nonparacompact space in ZFC.

The transitive, nonseparating, planar attractors whose existence has recently been established by Benedicks and Carleson all occur when the unstable manifold of a fixed hyperbolic saddle (the closure of which equals the attractor) is nearly tangent with the stable manifold of the saddle. Globally, these attractors are indecomposable continua. Unlike one-dimensional hyperbolic attractors, which are everywhere locally homeomorphic with the product of a Cantor set and an arc, these nonseparating attractors may have a very rich local structure as well.

Theorem: Suppose that F:R² --> R² is a C^oo diffeomorphism with fixed hyperbolic saddle p. Suppose also that a branch, W_+^u(p), of the unstable manifold of p has a (same-sided) nondegenerate tangency with the stable manifold of p and suppose that the logs of the eigenvalues of the Jacobian of F at p are independent over Q. Then in every neighborhood of every point of the closure of W_+^u(p) there is, in cl(W_+^u(p)), a homeomorphic copy of every continuum that can be expressed as an inverse limit of a sequence of unimodal bonding maps.

A polyadic space is a Hausdorff continuous image of some power of the 1-point compactification of a discrete space. We prove a Ramsey-like property for polyadic spaces which for Boolean spaces can be stated as: every uncountable clopen collection contains an uncountable subcollection which is either linked or disjoint. One corollary is that ($\alpha$$\kappa$)^$\omega$ is not a Universal preimage for Uniform Eberleins of weight at most $\kappa$, thus answering a question of Y. Benayamini, M. Rudin and M. Wage. Another consequence is that the property of being polyadic is not a regular closed hereditary property.

A longstanding problem is whether a compact metric continuum which is both homogeneous and hereditarily decomposable must be a simple closed curve. A proof will be given under the additional hypothesis that the continuum is locally connected. Some additional information about the (still unsolved) general case will be presented.

We will give a survey on the topological and quasi-conformal properties of the boundaries of Gromov-hyperbolic groups.

The discovery of Vassiliev invariants of knots and links suggested a natural object to investigate: the monoid of singular braids. One obtains this monoid by adding n-1 elementary singular braids to the n-string braid group. Motivated by the mathematics of Vassiliev invariants, we tried one day to "resolve singularities" in this monoid, much as Vassiliev resolved singularities in singular knots. This presented us with a strange map from the monoid of singlar braids to the integral group ring of the braid group, and to our surprise this map turned out to be a homomorphism! In fact, more is true: it's an isomorphism onto its image. The proof of this assertion is joint work with Antal Jarai, and it will be the subject of my talk.

Invariants of fibered knots in a rational homology sphere are defined using SU(n) representations of the knot group. In the case n=2, these specialize to Casson's invariant. These invariants detect irreducible SU(n) representations of the knot groups satisfying a monodromy condition along the longitude. The particular monodromy condition imposed is given by choosing a conjugacy class in the group SU(n), thus we get a family of invariants parametrized by SU(n) modulo conjugation. We study the effect on the invariants of varying the holonomy condition. With respect to a particular chamber structure on the parameter space, the invariants are constant within chambers and change by the product of two lower rank invariants when one moves between adjacent chambers.

Everyone has a few interesting problems which always come back to haunt them or perhaps, never seem to leave. Like a stray puppy, these problems might benefit by having other homes to visit. This will be a survey of a variety of such problems - including some history and partial results.

A Hausdorff continuum X is said to have the property of Kelley if for any subcontinuum K of X, any point p in K, and any open neighborhood U of K in C(X) (the hyperspace of all nonempty subcontinua of X) there is a neighborhood V of p in X such that for any q in V there is a continuum L satisfying q in L in U.

This property was defined for metric continua by J. L. Kelley and named Property (3.2) in J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52(1942), p. 22--36. Kelley has shown that continua with this property have contractible hyperspaces. Then basic properties of the property of Kelley were studied in R. W. Wardle, On a property of J. L. Kelley, Houston J. Math. 3(1977), p. 291--299. In particular, he has shown that homogeneous continua have the property of Kelley. Here we construct an example showing that this is not true for Hausdorff continua.

We use an example by S. B. Nadler, Jr. of a continuum Y such that Y has, while Y × Y does not have, the property of Kelley. A homogeneous Hausdorff continuum X is constructed such that it can be mapped onto Y by a monotone map. Since monotone (even confluent) maps preserve the property of Kelley, we conclude that X × X is a homogeneous continuum that can be mapped onto Y × Y by a monotone map, and thus X × X does not have the property of Kelley.

A dendroid means an arcwise connected and hereditarily unicoherent continuum. A dendroid which is locally connected is a dendrite. Let $\Cal M$ be a class of mappings such that all homeomorphisms are in $\Cal M$ and the composition of any two mappings in $\Cal M$ is also in $\Cal M$. A continuum X is said to be $\Cal M$-homogeneous provided that for every two points p and q of X there exists a surjective mapping f: X --> X such that f(p)=q and f is in $\Cal M$. We take as $\Cal M$ the class of monotone mappings (i.e., with connected point inverses) between continua. This class is contained in the class of confluent mappings (i.e. such f: X --> Y that if Q is a subcontinuum of the range Y and C is a component of f^-1(Q), then f(C)=Q).

QUESTION 1. What dendrites (dendroids) are monotone homogeneous?

It is known that a dendrite is monotone homogeneous if and only if it is confluent homogeneous. Is this equivalence valid for dendroids? H. Kato, [On problems of H. Cook, Topology Appl. 26 (1987), 219- 228] proved that the standard universal dendrite D₃ of order 3 (i.e. such that all its ramification points are of order 3 and the set of these points is dense in the dendrite) is monotone homogeneous.

THEOREM 1. If there exists a monotone mapping from a dendrite X onto D₃, then X is monotone homogeneous.

COROLLARY. If a dendrite has a dense set of its ramification points, then it is monotone homogeneous.

The converse is not true since there is a monotone homogeneous dendrite L₀ such that all its ramification points are of order 3 and the set of these points is nowhere dense (J. J. Charatonik, Monotone mappings of universal dendrites, Topology Appl. 38 (1991), 163-187).

THEOREM 2. If a dendrite contains the dendrite L₀, then it is monotone homogeneous.

QUESTION 2. Is the converse to the above implication true?

A dendroid X is said to be smooth if there is a point p in X such that for every point x in X and for every sequence {x_n} of points converging to x the sequence of arcs px_n converges to the arc px.

THEOREM 3. If a smooth dendroid is monotone homogeneous, then it is a dendrite.

QUESTION 3. Is smoothness an essential assumption in the above result?

Let M be a compact three-manifold with boundary containing k (greater than or equal to 1) tori. Assume that the interior of M is homeomorphic to H³/$\Gamma$, where $\Gamma$ is a nonelementary, torsion-free geometrically finite Kleinian group with the property that each parabolic element of $\Gamma$ is conjugate into one of k rank two parabolic subgroups corresponding to the tori. We show that the interiors of most manifolds obtained from M by Dehn filling admit geometrically finite hyperbolic structures that are geometrically close to H³/$\Gamma$.

We use the above result to obtain a general recipe for constructing examples of convergent sequences of discrete, faithful representations $\phi$_n: $\Gamma \prime$ --> PSL(2,C) of a group $\Gamma \prime$ whose algebraic limits are properly contained in their geometric limits.

We prove that there are uncountably many distinct inverse limit spaces with infinitely renormalizable quadratic maps as sole bonding maps.

Classification of group actions on 2-manifolds is a classical question arising from the Nielsen realization problem. On an oriented, closed (i.e. compact and without boundary) 2-manifold, actions of a finite abelian group can be classified by their fixed point data and free cobordism classes (Neilsen, Edmonds). The main difficulties for studying group actions on a noncompact 2-manifold is that it may have infinite genus or infinitely many branch points. We introduce the end data and the type of a cluster end to classify the actions on oriented, noncompact 2-manifolds.

We are interested in the topological identification of spaces of measures on metrizable spaces. Our main result states that the space P_s(X) of probability measures with separable supports is homeomorphic to a Hilbert space H if and only if X is completely metrizable and dens(H) = dens(X).

This is a joint work with Katsuro Sakai.

For uniform frames the "entourage" definition, proposed by P.Fletcher and W.Hunsaker in 1991, is used in a slightly modified version.

A uniform frame (L,U) is called Cauchy complete if every Cauchy filter on it is convergent.

A Cauchy complete uniform frame (L^*,U^*) is said to be a Cauchy completion of the uniform frame (L,U) if there exits a dense surjection $\varphi$: (L^*,U^*) --> (L,U).

Theorem. Every uniform frame (L,U) possesses a Cauchy completion (L^*,U^*), unique up to an isomorphism and called standard Cauchy completion, together with a dense surjection $\varphi$: (L^*,U^*) --> (L,U), unique up to the composition with an automorphism and called standard dense surjection, such that:

a) when the uniform frame (L,U) is Cauchy complete, it is isomorphic with (L^*,U^*);

b) for any Cauchy completion (L',U') of (L,U) and for any dense surjection $\varphi$': (L',U') --> (L,U) there is a (unique) dense surjection $\psi$: (L',U') --> (L^*,U^*) such that $\varphi$' = $\varphi \circ \psi$;

c) in the spatial case, if the uniform frame (L,U) is generated by the T₀-uniform space (X,$\Cal U$), its standard Cauchy completion (L^*,U^*) is isomorphic with the uniform frame $({\Cal O} \tilde X,\tilde U) generated by the uniform completion $(\tilde X,\tilde {\Cal U})$ of (X,${\Cal U}$).

We continue the study of the structure of compact spaces in the presence of countable restrictions on the convergence structure. Our experience leads to the view that the technique of iterated small forcings and reflection (model theoretic) is an integral part of the topic. We will defend this thesis (not that there are a lot of antagonists, thankfully) by illustrating its use in the proof of some recent theorems.

Theorem. It is consistent that every compact separable radial space is Fréchet.

Theorem. It is consistent to have the value of the continuum large and to have that compact sequential spaces are pseudoradial.

This last result is motivated by Sapirovski's CH result that compact sequential spaces are pseudoradial. It is a joint result with Juhasz, Soukup, and Szentmiklossy.

Eilenberg Theorem [1936] Assume that X is a compact metric space and dimX < or = n, n=k+l+1 and f: A --> S^k is a continuous map of a closed subset A contained in X to a k-sphere. Then there exists an extension F: X-Z --> S^k for some compact set Z of dimZ < or = l.

By Kuratowski's notations, X \tau K means that for every partial map f: A --> K defined on a closed subset A contained in X there is an extension over X. It is well-known that the property X \tau Sⁿ is equivalent to the inequality dimX < or = n. Note that Sⁿ = S^k * S^l is the join of a k-sphere and an l-sphere (k+l+1=n).

Generalized Eilenberg Theorem [1993] Assume that X is a compact metric space with X \tau K * L and f: A --> K is a continuous map of a closed subset A contained in X. Then there exists an extension F: X-Z --> K for some compact set Z with Z \tau L.

In contrast to two and three dimensional oriented orbifolds, which all bound, there are many oriented four orbifolds which cannot bound any oriented orbifold. We determine what the singular sets in four dimensional orbifolds can look like and then which of these orbifolds can bound. We give a complete description of the group of cobordism classes of oriented four orbifolds.

We will discuss the problem of iterating forcing while preserving the Continuum Hypothesis. In particular, we will discuss the author's joint work with Roitman on proving the consistency of CH with no Ostaszewski spaces. If time permits, we will talk a bit about a strong negation of $\clubsuit$ that is compatible with CH.

We consider the concepts of rotation number and rotation vector for area preserving diffeomorphisms of surfaces and their applications. In the case that that the surface is an annulus A the rotation number for a point x in A represents an average rate at which the iterates of x rotate around the annulus. More generally the rotation vector takes values in the one dimensional homology of the surface and represents the average "homological motion'' of an orbit. There are two main results. The first is that if 0 is in the interior of the convex hull of the recurrent rotation vectors for an area preserving diffeomorphism f isotopic to the identity then f has a fixed point of positive index. The second result asserts that if f has a vanishing mean rotation vector then f has a fixed point of positive index. Applications include the result that an area preserving diffeomorphism of A which has at least one periodic point must in fact have infinitely many interior periodic points. This is a key step in the proof of the theorem that every smooth Riemannian metric on S² has infinitely many distinct closed geodesics. Another application is a new proof of the Arnold conjecture for area preserving diffeomorphisms closed oriented surfaces.

Let F be a free Kleinian group on r generators. It is well known that F can be described as an inifinite 2r regular tree $\cal T$ and its limit set $\Lambda$(F) = $\partial {\cal T}$ can be given by the appropriate subshift of finite type on 2r generators. One can put the standard graph metric on ${\cal T}$ which induces a Cantorian metric on $\partial {\cal T}$ or the hyperbolic metric on $\cal T$ which induces the standard metric on $\Lambda$(F) contained in S². We then discuss the notion of the Hausdorff dimension of the limit set and the critical exponent of the corresponding Poincaré series with respect to the two metrics. In particular we shall give a simple lower bound for the classical critical exponent of the Poincaré series.

A topological space X is said to be nearly metacompact provided every open cover of X has an open refinement that is point finite on some dense subset of X [HL]. We will discuss some properties of nearly metacompact spaces.

Theorem. A space X is nearly metacompact if and only if every monotone open cover of X has an open refinement point finite on some dense subset of X.

Theorem. Every nearly metacompact lob-space is meta-Lindelöf.

Theorem. Every space can be embedded as a closed subspace of a nearly metacompact space.

[HL] R. Heath and W. Lindgren, ``On generating non-orthocompact spaces'', Set-Theoretic Topology, Academic Press, 1977, 225-237.

Recently, J. Heath proved that every 2-crisp map is a 2-fold covering map and that every 2-fold covering map has a 2-crisp restriction to a subcontinuum. She then asked about the relationship if 2 is replaced by an integer k greater than 2. We show that every k-crisp map is a k-fold cover and, for every k greater than 2, we construct a k-fold covering map which does not have a k-crisp restriction to a subcontinuum. A map is k-crisp if the domain is a continuum and for each proper subcontinuum C of the image, the inverse of C consists of k disjoint continua, each of which is mapped homeomorphically onto C.

A quasi-measure is a regular non-subadditive measure defined on sets which are either open or closed. Non-trivial examples of quasi-measures will be constructed on spaces such as the annulus and the torus. The cohomology ring is a fundamental object in this construction.

It is an unsolved problem to characterize in terms of X when the space C_k(X) of continuous real-valued functions on X with the compact open topology is a Baire space. We define and study a property called the moving off property, and show that for a q-space X (e.g., for X locally compact or first-countable), C_k(X) is Baire if an only if X has the moving off property.

Axioms are given for a preconvexity space and certain consequences obtained. In particular, it is shown that in a very natural way, a preconvexity on a space yields an abstract convexity space in much the same manner as a proximity yields a topolological space.

It is shown that every continuum of weight $\aleph$₁ (or less) is a continuous image of the Cech-Stone remainder $[0,\infty) ^*$ of the half-line. It follows that under CH the space $[0,\infty) ^*$ is a universal continuum of weight $\aleph$ ($=\frak c$). We compare the proof to Parivichenko's proof that every compact space of weight $\aleph$₁ (or less) is a continuous image of $\omega$ ^*.

We complement the result by showing that under MA every continuum of weight less than $\frak c$ is a continuous image of $[0,\infty)^*$ and that in the Cohen model the long segment of length $\omega$₂ is not a continuous image of $[0,\infty) ^*$.

Let A and B be square matrices of nonnegative integers. Let $\sigma$ _A and $\sigma$ _B be the subshifts of finite type associated with A and B. Let F_A and F_A be the topological suspensions of $\sigma$ _A and $\sigma$ _B. Let f: F_A --> F_A be a flow equivalence map. Parry and Sullivan showed that such an f exists if and only if A and B are related by a finite sequence of simple matrix transformations. The Bowen-Franks group is a well-known invariant of these transformations.

The matrices A and B are associated with directed graphs, which in turn can be associated with scalars in an abstract tensor algebra. A vertex with i incoming edges and j outgoing edges is represented by a symmetric tensor with i downstairs indices and j upstairs indices. Then it turns out that the matrix transformations of Parry and Sullivan correspond precisely to the axioms of a commutative cocommutative bialgebra, where the tensor is thought of as an i-fold multiplication followed by a j-fold comultiplication. From this point of view the Bowen-Franks group is most naturally seen as a group bialgebra which may be easily generalized to obtain an invariant semigroup bialgebra.

The set of topological conjugacy maps of the form h: $\sigma$_A --> $\sigma$_A is precisely the set of reversible 1+1-dimensional cellular automata. It turns out that flow equivalence maps of the form f: F_A --> F_A are closely related to a simple generalization of reversible 1+1-dimensional cellular automata in which spacetime curvature is permitted. (I call such maps "combinatorial spacetimes.") Here the components of F_A are interpreted as the one-dimensional combinatorial spaces; the map f induces a map between components, so it describes how these spaces evolve.

Motivated by the problem of robot navigation, the following question arises: To what extent can the singularities of a Morse function be prescribed? In particular, can the Morse function vary continuously as a function of the singularities? I will give some examples of when it can and when it can't. The main theorem is that a family of Morse functions with 3 singularities on the 2-disk can't vary continuously as a function of the singularities. Such a family would induce a homomorphism from the braid group on 3 strands to the braid group on 2 strands. Enough of this homomorphism can be computed to prove that it doesn't exist. This work is joint with Morris W. Hirsch of U.C. Berkeley.

We use separation axioms and infinite cardinals to classify a variety of metrization theorems. Of special interest is a characterization of the uniform weight u(X) of a space in terms of a condition much like the Frink Metrization Theorem. Nagata's Double Sequence Theorem and the Alexandroff-Urysohn Theorem also play an important role in the classification scheme.

An open problem in link-homotopy of links in the three-sphere is classification using some version of peripheral structures on reduced groups of links. We seek cases where a particularly straightforward version, called pre-peripheral structures , achieves classification, and also conditions under which counterexamples may occur. By formulating pre-peripheral structures in terms of string links (or pure braids), we can capture the relationship between link-homotopy equivalence and equivalence of pre-peripheral structures in a commutative diagram. The classification question then reduces to whether a certain group homomorphism is injective. For appropriately chosen collections of links, the groups are free abelian, so the question further reduces whether a certain matrix (whose entries are polynomials in generalized linking numbers) has non-trivial nullspace. To find counterexamples, then, we must find integer solutions to systems of polynomial equations derived from these matrices.

We consider the dynamical system (M,S) where M is the orbit closure of a nonperiodic recurrent sequence of 0's and 1's (for example, the Morse sequence) and S is the shift map. The enveloping semigroup is E(M) = cl[ Sⁿ : n in Z], where the closure is taken in the topology of pointwise convergence. H. Furstenberg was the first to establish the existence of relationships between recurrence, IP sets, and idempotents in the enveloping semigroup, and the first author has proven that the closure of the set of idempotents coincides with the IP cluster points. In this paper the authors compute this set for (M,S) and shed light on other combinatorial properties of generalized Morse sequences.

We investigate a question of Waldhausen, which asks if the rank of the fundamental group of a 3-manifold differs from its Heegaard genus. Boileau and Zieschang produced examples of closed Seifert-fibred manifolds for which the Heegaard genus exceeds rank. It is still unknown if these two quantities ever differ for non-Seifert manifolds or bounded manifolds. For bounded manifolds, there is a sharper form of this question, the meridional generator conjecture. This conjecture asserts that meridional generators of the group correspond to bridges as properly interpreted. New results concerning knots and this conjecture will be presented.

If X is a topological space in a ground model V and P is any notion of forcing then X may also be considered as a topological space in the generic extension V^P with the topology generated by the ground model open sets as a basis. Thus, for any cardinal function f, we may then ask about the relationship between the value f(X) taken in V and the value f ^P(X) taken in V^P. Our aim here is to study this relationship for various cardinal functions, when P is a CCC or some special kind of CCC notion of forcing (e.g. P adds Cohen reals, is sigma-centered or sigma-linked, has property K, etc.).

In 1980, A. A. Gryzlov proved that every compact T₁-space of countable pseudocharacter has cardinality at most that of the continuum. We show that the relativization of Gryzlov's theorem, i.e., the statement one gets by replacing "compact" by "compact in a larger space", is false.

We use normed linear space arguments to obtain local results about the initial value problem:

Barge and Martin have shown that the inverse limit of any interval map may be realized as an attractor of a planar homeomorphism H in such a way that H restricted to the attractor is conjugate to the induced homeomorphism on the inverse limit. Since the inverse limit of any interval map is chainable, a logical question is: "given a chainable continuum X and a homeomorphism g: X --> X, is there an embedding $\Psi: X --> R² so that there exists a planar homeomorphism G such that $\Psi$(X) is an attractor for G and G|_$\Psi$(X) is conjugate to g?". An example showing that the conjecture is false is given. Moreover, it is shown that given any nondegenerate planar continuum $\Lambda$ and homeomorphism f: $\Lambda$ --> $\Lambda$, which is not the identity, there is an embedding $\phi$: $\Lambda$ --> R³ such that $\phi$ º f º $\phi$^-1 cannot be extended to all of R³.

We use Lagrangian dynamics to construct a model of fluid flow past an infinite bisequence of cylinders. Our fluid flow takes place in the plane. The model is investigated first numerically, and then simpler models of the Poincaré return map (based on our observations of behavior of the flow) are explored from a rigorous and topological standpoint.

(This is joint work with Celso Grebogi, Miguel Sanjuan, and James Yorke, all of the University of Maryland.)

How large is a basin of attraction? The question can be interpreted in a variety of ways. Here, we present an elegant result for single-variable complex rational functions which, in certain cases, yields a method for determining the exact distance from an attracting fixed point to the Julia set. Such a distance is a bound on the amount of error tolerable in initially approximating an attracting fixed point of a complex rational function.

In this work, we compute the first homology groups of the mapping class groups of connected closed non-orientable surfaces. It turns out that they are isomorphic to Z₂ if the genus of the surface is at least seven. We conclude that the group of isometries of a vector space over Z₂ with the Euclidean symmetric bilinear form is perfect.

A minimal set for a flow $\Phi$: R × M --> M is a set that is invariant, closed, non-empty, and minimal in that respect. T. Inaba proved that there is a flow with no minimal set on a 2-dimensional surface of infinite genus. To answer a question of P.A.Schweitzer, we show that not every flow on R³ has a minimal set.

A quasi-measure on a compact Hausdorff space X is a finitely additive "measure" on $\cal A$ (the collection of subsets of X which are either open or closed) which is not required to be sub-additive. Thus, if µ is a quasi-measure on X that is not a measure, then there are closed sets C₁ and C₂ such that µ(C₁) = 0 = µ(C₂), but µ(C₁ U C₂) is not 0.

We prove that if µ is a quasi-measure on a compact X, then µ is countably additive. I.e., if {A_n} is contained in $\cal A$ and U{A_n} is an element of $\cal A$, then µ(UA_n) = $\sum$ µ(A_n). As a consequence, we obtain a correspondence between the collection of 0--1 quasi-measures on X and a subset of the collection of maximal linked families of closed connected subsets of X.

We discuss several recent results as well as several questions involving hereditarily indecomposable continua. Topics covered include the following:

1. hyperspaces
   
2. characterizations of continua
   
3. hereditary equivalence
   
4. retracts
   

Simplicial dynamics occurs very often in topological dynamics. That is the case, for instance, for homeomorphisms or flows when a Markov partition is available. In turn, the simplicial structure if it is associated with a finiteness result is very usefull for some decision problems such as: - Is a surface homeomorphism isotopic to a pseudo-Anosov? - Are two elements of the mapping class group of a surface conjugated? - Does a braid admits a destabilization? - Does a hyperbolic flow admits a global cross section? In this talk I will discuss several aspects of simplicial dynamics which has been used recently to solve such decision problems.

We conjecture that there is a function, $\tau$, from the nonnegative integers to the nonnegative integers having the following property: If K is a knot in the 3-sphere which admits a Dehn surgery containing an essential surface of genus g then the exterior of K contains a properly embedded essential surface of genus at most g with at most $\tau$(g) boundary components. Note that $\tau$ is independent of K. We prove the existence of $\tau$ in the case where the surgery slope is non-integral.

This talk reports on recent joint work with Hal Bennett. In his 1965 paper [On $\sigma$-discreteness and Borel isomorphism, Amer J.Math., 85 (1963), 655-666], A.H. Stone described a metric space X with cardinality $\omega$₁ such that X is not $\sigma$-discrete and every separable subset of X is countable. The space X is a special subset of D^$\omega$, where D is an uncountable discrete space with cardinality $\omega$₁. Let Y be the closure of X in D^$\omega$. Then Y is completely metrizable and, according to a 1965 theorem of Herrlich, there is some linear ordering of Y whose open-interval topology coincides with the topology that Y inherits from D^$\omega$. That allows us to construct a linearly ordered space E that is non-metrizable, first-countable, paracompact, perfect, Cech-complete, and has a dense subspace that is the union of countably many closed discrete subspaces of E. The purpose of this talk is to discuss E and its relationships to two older problems concerning $\sigma$-minimal bases and small diagonals.

1) An open problem asks whether a compact linearly ordered topological space Z must be metrizable if every subspace of Z has a $\sigma$-minimal base (in the sense of Aull) for its relative topology. Such a space Z must be first-countable and hereditarily paracompact, and D. Burke pointed out every subspace of Z must contain a dense metrizable subspace. The space E above is not compact, of course, but it is Cech complete, and one can show that every subspace of E has a $\sigma$-minimal base. Therefore, while the question for compact ordered spaces remains unresolved, the space E places a limit on how far a positive result could be generalized. Even if it turns out to be true that a compact LOTS having $\sigma$-minimal bases hereditarily must be metrizable, that result could not be generalized beyond the class of locally compact ordered spaces.

2) Recently, Juhasz and Szentmiklossy [Proc. Amer. Math. Soc., 116 (1992), 1153-60] proved that (*) under CH, a compact Hausdorff space with a small diagonal (in the sense of Husek) must be metrizable. One would expect that their result could be proved for the successively larger classes of paracompact, locally compact spaces, then paracompact Cech-complete spaces, and finally for paracompact p- spaces. The space E shows that is not the case: the generalization must stop with paracompact, locally compact spaces. In addition, some years ago, van Douwen and I proved that (**) a compact (or even Lindelöf) linearly ordered topological space with a small diagonal must be metrizable, and the space E shows that (**) also resists generalization. Indeed, the space E has an even stronger property called an "H-diagonal": for any regular uncountable cardinal $\kappa$ < |E|, if T is a subset of E² - $\Delta$(E) has cardinality $\kappa$, then there is an open neighborhood W of $\Delta$(E) in E² such that |T-W| = $\kappa$.

Every area and orientation preserving homeomorphism of the annulus that rotates the boundary circles in opposite directions must contain two fixed points. This is the Poincaré-Birkhoff Theorem. Similar results exist for the open annulus, basin boundaries, and many plane continua. These, including a theorem for irreducible plane separating continua, are discussed.

Let $\phi$ be a diffeomorphism of the closed unit 2-disc D². A ball B_$\delta$(x) centered at a point x of $\partial$D² and of radius $\delta$ > 0 is called + (resp. -) -twisted if, writing $\phi$ as (r,$\theta$) --> (R,$\Theta$) in the polar coordinates of D², d$\Theta$/dr is nonnegative (resp. nonpositive) on B_$\delta$(x) $\cap$ Int(D²) and nonvanishing on $B_$\delta$/8(x) $\cap$ Int(D²). Our main result is the following.

Theorem 1. Let $\phi$ be an Euclidian area preserving diffeomorphism of D² keeping the points of $\partial$D² fixed. If there exist both + and - -twisted balls of radius $\delta$ and if $\phi$ is $\delta$/8 near to the identity in the C¹ topology, then the fixed point set of $\phi$ in Int(D²) is nonempty and nonconnected.

Let us show how Theorem 1 is obtained. Let $\cal H$ be the space of smooth maps H:[0,1]× R²ⁿ --> R such that Supp(H) = Cl($\bigcup$_tSupp(H_t)) is compact, where we denote H_t(x) = H(t,x). Given H in $\cal H$, let $\phi$^t_H (with t in [0,1]) be the one parameter family of diffeomorphisms of R²ⁿ determined by the equations d$\phi$^t_H/dt = J$\nabla$ H_t º $\phi$^t_H and $\phi$_H⁰ = id, where J is the standard almost complex structure and $\nabla$ stands for the gradient. Given a fixed point x of $\phi$ = $\phi$¹_H, define the action A(x;H) by

Theorem 2. If there exists an open ball B_$\epsilon$(y) such that H is nonnegative (resp. nonpositive) on [0,1] × B_$\epsilon$(y) and H_t(y) is positive (resp. negative) for some t, and if sup|$\nabla$ H| < 4 $\epsilon$, then there exists a fixed point x of $\phi$ such that A(x;$\phi$) is negative (resp. positive).

[HZ] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, 1994 Birkhäuser, Basel

In recent work, Gabai has proved that if it can be shown that a rather large and well-situated (with respect to itself) solid tube (around some geodesic) exists in each and every hyperbolic 3-manifold, then homotopy hyperbolic 3-manifolds are actually hyperbolic. In joint work, Gabai, N. Thurston, and I have developed a powerful approach to constructing such solid tubes. Our approach uses (rigorously) computers. Various aspects of our method will be discussed, and the present status of our project will be revealed.

We will discuss various covering properties of sets of reals which are due to Rothberger, Menger, Hurewicz, Gerlits-Nagy, and Borel. In particular, we will discuss a conjecture of Hurewicz which is analogous to the Borel conjecture about strong measure zero sets of reals. A set of reals X has the Hurewicz property, if whenever we are give a sequence <${\Cal U}$_n: n in $\omega$> of open covers of X we can choose <${\Cal U}$_n in <[${\Cal U}$_n]^$\omega$: n in $\omega$> such that every element of X is in all but finitely many $\cup {\Cal U}_n$. It easy to see that if X is $\sigma$-compact (the countable union of compact sets), then X has the Hurewicz property. The Hurewicz conjecture is that the converse is true.

A rather natural question to ask is whether or not Sⁿ is homeomorphic to S^m for some distinct n,m in N. The answer to this question was shown to be no in ZFC (D. Burke, D. Lutzer) but until recently it was not even known whether there exists an uncountable subspace X of S with X² homeomorphic to X³. Along with discussing other interesting properties of S, I will prove in ZFC that there are no uncountable subspaces X of S such that an open neighborhood in Xⁿ⁺¹ embeds in Sⁿ.

In this paper we use the concepts of link diagram colorability and branched cyclic covering spaces to study links under the piecewise linear isotopy equivalence relation (non-ambient isotopy; local knots are allowed to be tied and removed). We also show that the "nullity corrected" Goeritz matrix of a link, which presents the first homology group of the two fold cyclic branched covering space, determines all possible diagram colorings. However, unlike the case for knots, it does not determine all maps of the link group onto generalized dihedral groups.

Let $\{X\sb\alpha:\alpha\in A\}$ be a family of spaces labelled by elements of a directed set $(A,\le)$.

Let I be the family of all subsets $$ of the product space $X=\Pi\sb{\alpha\in A}\,X\sb\alpha$ such that there exist continuous functions $f\sb\beta\sp\alpha:X\sb\alpha\to X\sb\beta$, $\beta\le\alpha\in A$, such that the following two conditions are satisfied:

\item{(1)} $f\sb\alpha\sp\alpha={\rm id}\sb{X\sb\alpha}$ for each $\alpha$, and $f\sb\gamma\sp\beta\circ f\sb\beta\sp\alpha=f\sb\gamma\sp\alpha$ when $\gamma\le\beta\le\alpha$, and

\item{(2)} $\displaystyle =\lim\limits\sb\leftarrow(X\sb\alpha, f\sb\beta\sp\alpha,A)$.

The case when all spaces $X\sb\alpha$ are compact metrizable and I is treated as a subspace of the hyperspace $2\sp X$ is studied.

This is the joint work with H.M.Tuncali and E.D.Tymchatyn.

We consider random dynamical systems, which model dymamics influenced by noise. Stationary random variables (or equivalently, stationary random Dirac measures) are a generalisation of fixed points of deterministic dynamical systems. We investigate the influence of small random perturbations on fixed points of deterministic maps. Using results from algebraic ergodic theory we give examples of fixed points which are structurally stable w.r.t.\ perturbations in the class of deterministic maps but not structurally stable w.r.t.\ stochastic perturbations. In particular, we show that topological fixed point theorems like the Brouwer fixed point theorem or theorems based on the Conley index theory cannot be generalized to the case of random dynamical systems.

The length of a group G is the least ordinal $\alpha$ such G_$\alpha$ = G_$\alpha$+1 where G_$\alpha$ is the $\alpha$^th term in the transfinite lower central series of G. We establish connections between the length of compact three manifold groups and various central open problems such as the Parafree conjecture, the link slice problem, and the four dimensional topological surgery conjecture. We prove prove the length of all surface groups and most Fuchsian groups is at most $\omega$. Similar results hold for Seifert fibered three manifold groups. Our main result is the existence of hyperbolic three manifold groups of length at least 3$\omega$

We will study the characterization of the spaces mentioned in the title.

3 and 4 dimensional maximal geometries in the sense of Thurston, are easily understood using the mechanichs of Cartan triples. We discuss a two step procedure on how to get a high degree of symmetry rigid structure on a three manifold starting with a trivialization of the tangent bundle. In particular one shows that the moduli space of local isometry classes of c.h. metrics with a spherical homogeneous model is infinite dimensional.

Zsilinszky asked Watson, "Is there a space X such that X² is Baire and X has a dense Baire subset Y such that X × Y is not Baire?" An affirmative answer is found by modifying an example from W.G. Fleissner and K. Kunen, Barely Baire spaces, Fund. Math. 101 (1978) pp.229-240. The question can be generalized to ask that Y dense in X has certain Baire products Xⁿ × Yⁿ. The construction generalizes to yield a characterization of such questions that can be satisfied.

Periodically forced planar oscillators are often studied by varying the two parameters of forcing amplitude and forcing frequency. For low forcing amplitudes, the study of the essential oscillator dynamics can be reduced to the study of families of circle maps. The primary features of the resulting parameter plane bifurcation diagrams are "(Arnold) resonance horns" emanating from zero forcing amplitude. Each horn is characterized by the existence of a periodic orbit with a certain period and rotation number. In this paper we investigate divisions of these horns into subregions -- each subregion corresponding to maps having a constant number of periodic orbits. Subregions having more than the "usual" two periodic orbits can be interpreted as "folds" in the corresponding surface of fixed points in the phase × parameter space. Some of the resulting bifurcation pictures in the parameter plane appear in shapes we call "Arnold flames". This study leads to a simple method for constructing families of maps with bifurcation features such as flames and swallowtails. Results apply both to circle maps and forced oscillator maps.

Which locally-connected continua are the k-to-1 image of a dendrite? We discuss some recent results concerning necessary and sufficient conditions for a continuum to be such an image.

An example is given of a space of uncountable tightness whose product with any suborderable space is normal. The motivation for this is a result in a paper by M.E. Rudin and me stating that the product of a suborderable space with a countably tight $\sigma$- compact space is normal.

Let M be a finite volume hyperbolic 3-manifold whose boundary consists of a finite number (non-zero) of incompressible tori. An orientable closed surface S immersed in M is called essential if it is incompressible and not boundary parallel. A question dating back to Waldhausen and discussed in various contexts by Thurston is the problem of the extent to which irreducible 3-manifolds with infinite fundamental group must contain surface groups. Here we show:

Theorem: With M as above. Then $\pi$₁(M) has a finite index subgroup which maps onto a free group of rank 2.

In fact we show the following:

Theorem: M as above. Then M contains an immersed essential closed surface which lifts to an embedded non-separating surface in a finite cover.

It is a consequence of the proofs of these theorems that "non-peripheral homology" can be made arbitrarily large by passing to finite covers of M.

This is joint work with D. Cooper and D. D. Long.

A paper, written jointly with Evgenij V. Scepin, will be presented. Its major result is a proof of the Lipschitz case of the classical Hilbert-Smith Conjecture, to the effect that the p-adic integers cannot act freely with Lipschitz maps on any closed manifold.

An outline of the proof of the consistency of "CH + there are no Ostaszewski spaces," joint work with Todd Eisworth.

Approximate (inverse) systems of compacta have been useful in the study of covering dimension, dim, and cohomological dimension over an abelian group G, dim_G. 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 prove that if G is an abelian group, a compactum X is the limit of an approximate system of compacta X_a, n in N, and dim_G X_a is less than or equal to n for each a, then dim_G X is less than or equal to n.

Ben Fitzpatrick and Zhou Hao Xuan asked if there is a countable dense homogeneous metric space which is not completely metrizable. They then gave an example using the continuum hypothesis. Also, an example has been given under Martin's Axiom. We will give an example in ZFC.

We prove the following theorem.

1.1 Theorem. Let n be a nonnegative integer, X = {X_i ,f_i,i+1 ,N} be an inverse sequence of metric spaces X_i, with limit X, G be a finitely generated abelian group, and suppose that dim_G X_i is less than or equal to n for each i in N. Then dim_G X is less than or equal to n.

The proof defines a metric space Z with covering dimension less than or equal to n and a proper map $\pi$ from Z to X with suitable fibers.

In early attempts to classify manifolds, topologists studied and compared possible triangulations. They thereby succeeded in classifying 1- and 2-manifolds, but were eventually satisfied that this approach was not terribly useful for manifolds of dimension 4 and higher. For 3-manifolds, the use of triangulations (or Heegaard splittings as they're called in the trade) was at first promising, eventually disappointing, but has recently attracted attention again. Modern advances have given us a clearer picture of how Heegaard splittings behave and several new 3-manifold techniques have been based on their properties. We'll discuss these new insights in general and, more specifically, new techniques for comparing Heegaard splittings of the same 3-manifold.

Up to homeomorphism there is a unique 4-manifold that is homotopy equivalent but not homeomorphic to the complex projective plane; i.e., the Chern manifold. Results of Kwasik, Wilczynski and others during the nineteen eighties showed that this manifold admits a large assortment of odd order cyclic group actions. The main result of this work is that the Chern manifold admits no nontrivial involutions that induce the identity on homology; it follows that the cyclic group of order 2 is the only finite 2-group that can act effectively on this manifold. The proof uses a mixture of techniques from cohomological fixed point theory and controlled surgery theory; this is joint work with Slawomir Kwasik.

It is known that the disk is homogeneous with respect to open maps. We show that the disk is not homogeneous with respect to monotone maps. We then discuss how this result can be generalized to other planar manifolds with boundary. Finally we show a non-compact planar manifold with boundary that is homogeneous with respect to monotone open maps.

One of the most useful ordinal invariants for sequential spaces is sequential order. Many questions about the behavior of this invariant arise naturally. Among them is the question about the nontriviality of sequential order on several classes of sequential spaces. In 1981 P. Nyikos asked if sequential order could take on nontrivial values on the class of topological groups. We propose under CH a construction of a topological group which answers this question affirmatively.

This talk contains some recent results on countable metacompactness in subspaces of products of ordinals. In particular, we show that $\kappa$² is hereditarily countably metacompact for any ordinal $\kappa$. We also discuss the failure of countable metacompactness in certain subspaces of infinite products of ordinals.

We study the problem of approximating planar rotations (about the origin)

A scheme S associates to every $\alpha$ a $\beta$. That is, if K is the space of rotations and L is the space of one-to-one, onto lattice maps, then

Suppose M is a closed, simply-connected smooth 4-manifold. Then M has a handlebody structure in which a subset of the 2-handles algebraically cancels the 1-handles and a disjoint subset of the 2-handles dually algebraically cancels the 3-handles. As a corollary one gets an improved version of the $\Lambda$-splitting theorem of M. Freedman and L. Taylor. If q_M = $\lambda₁ \oplus \lambda₂$ is any decomposition of the intersection form of M then there is a decomposition M = M₁ \cup_$\Sigma$ M₂, where $\Sigma$ is a homology 3-sphere, M₁ and M₂ are smooth and simply connected, and the intersection form of M_i is isomorphic to $\lambda$_i. Also one gets an improved version of the splitting theorem of Curtis-Freedman-Hsiang-Matveev-Stong. Suppose M₁ and M₂ are homeomorphic simply-connected smooth 4-manifolds. Then there are decompositions

Smale flows form a class of structurely stable flows. We define a Lorenz-Smale flow to be a Smale flow on the 3-sphere with three basic (invariant) sets: an attracting closed orbit A, a repelling closed orbit R, and a 'Lorenz-type' saddle set L. This forces A and R to form either a Hopf link or a trefoil with meridian link. Further at most one of the two branches of L may be knotted, and then its knot types are limited to be torus knots.

(Editor's Note: The author notes that a preview of this talk will be available on the web page: http://www.math.nwu.edu/~mcs/ . This site will be delinked from this abstract after the conference.)

Periodic prime noncompact knots properly embedded in the infinite cylindar I² × R where I is the unit interval, are ambiently isotopic iff their diagrams are equivalent by a finite sequence of Reidemeister moves.

We give sufficient conditions for a Tychonoff space to have a Hausdorff connectification. We show, for example, that every nearly realcompact Tychonoff space without compact open subspaces and with no more than exp ( c ) disconnections has a Hausdorff connectification. Our results give some partial answers to questions of Watson and Wilson.

We study the question whether compact subsets C_p(X) have countable tightness where X is a Lindelöf space. A new class of spaces X is defined for which all compact subsets of C_p(X) have countable tightness. It is open whether this class includes all Lindelöf spaces.

A subspace Y of a space X is called regular in X if each element y of Y and each closed (in X) subset of X that does not contain y can be separated by open (in X) sets. Compact Hausdorff spaces are characterized as Hausdorff spaces which are regular in every larger Hausdorff space.

A central problem in 3-dimensional topology is to classify 3-dimensional spaces. That is, given a collection of tetrahedra, together with instructions on how to glue their faces together, we'd like to be able to decide algorithmically what the resulting 3-manifold is. Until recently there was no algorithm even to decide whether or not the resulting manifold was the 3-sphere. This is called the recognition problem for the 3-sphere. I'll discuss a solution to the recognition problem from the point of view of knot theory.

The main result is that there exist countably many ambiently inequivalent unknotted tame embeddings of the pseudo-circle in 3-space. This step removes the "PL" restriction of Part I.