Topology

Faculty

The full time employees in the past few years have been: Kazimierz Alster, Józef Krasinkiewicz, Wiesław Olędzki, Stanisław Spież and Henryk Toruńczyk. Moreover, within the last 3 years the following topologists have been or are currently employed for a period varying from 6 months to 2 years: T. Dobrowolski, K. Gęba, G. Graff, M. Izydorek, T. Januszkiewicz, J. Jezierski, D. Kołodziejczyk, T. Koźniewski, S. Nowak, R. Mańka, J. Świątkowski, A. Tralle, P. Traczyk.

Below are the descriptions of the main research results or research areas of the full time employees and the list of some of the papers published (either containing the results discussed or completed within the last 3 years).

Kazimierz Alster

My research area belongs to General Topology. I have been interested in studying the properties of a space which can be expressed in terms of its coverings. These include: the Hurewicz property, the Eberlein compactness, the Fréchet property, paracompactness, and the Lindelöf property of Cartesian products. The study of these properties often leads to problems close to infinite combinatorics.

Józef Krasinkiewicz

My main results include:

1. Showing in [11] that if every map of a compactum X into R²ⁿ can be approximated by an embedding then dim (X×X) < 2n. (The reverse implication and other extensions have subsequently been obtained by many authors, primarily by S. Spież and by H. Toruńczyk.)
2. Proving, jointly with Z. Karno [9], that the basic property dim(X×X) < 2 dim(X) celebrated Boltyanskii-Kodama compacta and of their natural n-dimensional counterparts can be established using the above result by studying the set of all mappings of these spaces into R²ⁿ. (This research grew out from a joint paper with K. Lorentz in Bull. Polish Acad. Sci. 36 (1988), 397-402.)
3. Proving that mappings of a compact metric space into manifolds can be approximated by mappings whose fibers are hereditarily indecomposable, and that in fact the set of such maps is a dense G_δ (see [12]).
4. Extending the notion of an essential mapping into spheres to the case when the target space is a product of manifolds. This new notion has been applied (see [10]) to construct spaces with peculiar properties (e.g.: there exists a nondegenerate continuum whose each subset of positive dimension admits an essential map onto each sphere) and to proving decomposition theorems (e.g.: if the Hilbert cube Q is expressed as a countable union of sets, then one of them contains a connected non-trivial subset, which strengthens Hurewicz's result on decompositions of Q into zero-dimensional sets).

Stanisław Spież

My major areas of research are shape theory (and indirectly homotopy theory), dimension theory, theory of embeddings and game theory.

Several of my papers in shape theory are devoted to studying movable spaces (their role is similar to that of CW-complexes in homotopy theory) and deformation dimension (which corresponds to the homotopical dimension). Some of my results in that area are related to the classical Whitehead and Hurewicz theorems in homotopy theory. Also I investigated the possibility of representing the strong shape category in the homotopy category.

Another area of my research is related to the following questions: "When can a pair of mappings of compact metric spaces X and Y into Rⁿ be approximated by mappings with disjoint images, and also when can a map X→Rⁿ be approximated by embeddings?'' Since the 1930's the standard answer to the latter has been "It suffices that 2 dim X < n" turns out that it is sufficient that dim(X×X) < n. Some other papers of mine concern the first question (which is more general).

Also I was interested in the questions of embedding polyhedra into Euclidean spaces, which were related to the van Kampen and Haefliger-Weber theorems.

Recently I am also involved in research in game theory. Some results on the existence of equilibria in a class of games can be proved by using topological tools, such as coincidence theorems of Borsuk-Ulam type.

Several of the above results were obtained in collaboration with the following mathematicians: B. Günther, S. Nowak, J. Segal, R. Simon, A. Skopenkov and H. Toruńczyk.

Henryk Toruńczyk

Major part of my research concerned topological properties of infinite-dimensional spaces, such as the Hilbert cube or Banach spaces. I consider the following my main results:

• developing a method of constructing smooth partitions of unity on Banach spaces in the absence of separability [20];
• proving that a product of an absolute retract with an appropriate normed linear space becomes homeomorphic to that space [21];
• proving, simultaneously with S. Ferry, that the homeomorphism group of a Hilbert cube manifold is a manifold [22];
• characterizing infinite-dimensional manifolds topologically (see R. D. Edwards' article in SLN 770, 278-302). As a consequence it turned out that infinite-dimensional Banach spaces of the same weight are homeomorphic;
• examining, jointly with S. Spież, when mappings X, Y→R^k can be ε-approximated by mappings with disjoint images [18];
• establishing, jointly with R. Simon and S. Spież, the existence of equilibria in a class of infinitely repeated games. (The proof in [19] depended on developing an appropriate topological aparatus.)

Selected papers

1. K. Alster, Some remarks on Eberlein compacts, Fund. Math. 104 (1979), 43-46.
2. K. Alster, On Michael's problem concerning the Lindelöf property in the Cartesian products, Fund. Math. 121 (1984), 149-167.
3. K. Alster, On the product of a Lindelöf space with the space of irrationals under Martin's Axiom, Proc. Amer. Math. Soc. 110 (1990), 543-547.
4. K. Alster, On the Cartesian products of Lindelöf spaces with one factor hereditarily Lindelöf, Proc. Amer. Math. Soc. 116 (1992), 207-212.
5. K. Alster (with R. Pol), A remark on Kurepa's topology on Aronszajn trees, Houston J. Math. 18 (1992), 409-415.
6. K. Gęba, M. Izydorek and A. Pruszko, The Conley index in Hilbert spaces, Studia Math. 134 (1999), 217-233.
7. M. Izydorek (with Z. Dzedzej), The Borsuk-Ulam theorem for differentiable maps, in: Differential Inclusions and Optimization Control, Lecture Notes in Nonlinear Anal., 1998, 139-143.
8. M. Izydorek (with Z. Dzedzej and A. Vidal), On a theorem of Tveberg, Topol. Methods Nonlinear Anal., to appear.
9. J. Krasinkiewicz (with Z. Karno), On some famous examples in dimension theory, Fund. Math. 134 (1990), 213-220.
10. J. Krasinkiewicz, Essential mappings onto products of manifolds, in: Banach Center Publ. 18, PWN, 1986, 377-406.
11. J. Krasinkiewicz, Imbeddings into Rⁿ and the dimension of products, Fund. Math. 133 (1989), 247-253.
12. J. Krasinkiewicz, On mappings with hereditarily indecomposable fibers, Bull. Polish Acad. Sci. 44 (1996), 147-156.
13. S. Spież (with B. Günther and J. Segal), Strong shape of uniform spaces, Topology Appl. 49 (1993), 237-249.
14. S. Spież (with J. Segal and A. Skopenkov), Embeddings of polyhedra in R^m and the deleted product obstruction, Topology Appl. 85 (1998), 335-344.
15. S. Spież (with J. Segal), Quasi-embeddings and embeddings of polyhedra in R^m, Topology Appl. 45 (1992), 275-282.
16. S. Spież, Imbeddings in R^2m of m-dimensional compacta with dim(X×X) < 2m , Fund. Math. 134 (1990), 103-113.
17. S. Spież, On pairs of compacta with dim(X×Y) < dim X + dim Y , Fund. Math. 135 (1990), 213-222.
18. S. Spież and H. Toruńczyk, Moving compacta in R^m apart, Topology Appl. 41 (1991), 193-204.
19. S. Spież, H. Toruńczyk (and R. Simon), The existence of equilibria in certain games, separation for families of convex functions and a theorem of Borsuk-Ulam type, Israel J. Math. 92 (1995), 1-21.
20. H. Toruńczyk, Smooth partitions of unity on some non-separable Banach spaces, Studia Math. 46 (1973), 43-51.
21. H. Toruńczyk, Absolute retracts as factors of normed linear spaces, Fund. Math. 86 (1974), 53-67.
22. H. Toruńczyk, Homeomorphism groups of compact Hlbert cube manifolds which are manifolds, Bull. Acad. Polon. Sci. 25 (1977), 401-408.
23. H. Toruńczyk (with J. E. West), Fibrations and bundles with Hilbert cube manifold fibers, Mem. Amer. Math. Soc. 406 (1989).
24. A. Tralle (with J. Oprea), Symplectic Manifolds with no Kaehler structure, Lecture Notes in Math. 1661, Springer, 1997.
25. A. Tralle, Rational models of solvmanifolds with Kaehlerian structures, Revista Math. 10 (1997), 157-176.
26. A. Tralle (with W. Andrzejewski), Flat bundles and formality, Ann. Polon. Math. 65 (1997), 105-176.
27. A. Tralle, Compact symplectic and Kaehler solvmanifolds which are not completely solvable, Colloq. Math. 73 (1997), 261-283.
28. A. Tralle (with J. Oprea), Massey products and the examples of McDuff, to appear.