Homotopic dominations of polyhedra

Aim of project
We consider here some natural, easily formulated, old but still unsolved problems on homotopic dominations of polyhedra. We concentrate on questions concerning the cardinality of the class of spaces homotopically dominated by a polyhedron (up to finiteness). We also study an order in the class of polyhedra homotopically dominated by a given polyhedron determined by the relation of domination.

The starting point of our investigations was the problem posed in 1968 by a great Polish topologist Karol Borsuk: ”Does every polyhedron dominate homotopically only fi nitely many different homotopy types?”

Unexpectedly, the answer is negative. On the other hand, for example, polyhedra with finite fundamental groups dominate homotopically only finitely many different homotopic types. We will continue that previous work and try to find other classes of polyhedra dominating homotopically only finitely many different homotopic types.

The problems studied in this project include, among others, a similar question concerning homotopy decompositions of polyhedra into Cartesian products (K. Borsuk, 1971): ”Does there exist a polyhedron with infinitely many different Cartesian factors in the homotopy category of CW-complexes?” The aim of this project is also to answer some other open questions on this and related subjects. Some of them were posed in the papers and books of K. Borsuk or included in the known list ”Open Problems in Topology” published regularly by Elsevier. The results will be used to complete the habilitation dissertation of the supervisor of the project. They will also be published in international mathematical journals and presented at
conferences.

Expected results
We expect to answer (or partially answer) some open questions related to this subject. Expected results include, among others, a proof that every polyhedron with a virtuallypolycyclic fundamental group has a finite depth. We will distinguish some classes of polyhedra in which every homotopic domination of a polyhedron over itself is a homotopic equivalence. We also expect to show that every polyhedron with a nilpotent fundamental group decomposes only in fi nitely many ways into a Cartesian product with the second factor S1 (in the homotopy category of CW-complexes).