Homotopy dominations of polyhedra
by
Danuta Kolodziejczyk
Warsaw University of Technology, Poland

In this talk we will present and discuss a solution to the problem: Does every polyhedron dominate only finitely many different homotopy types?

This question was stated by K. Borsuk in 1968, at the Topological Conference in Herceg-Novi, in an equivalent, shape formulation (see also Russ. Math. Surv. 34:6, 1979; Studies in Topology, Acad. Press, 1975).

By a polyhedron we always mean a finite one.

We showed that the answer to the Borsuk's question was negative. Moreover, the examples are not rare: for every non-abelian poly-Z-group G there exists a polyhedron with fundamental group isomorphic to G dominating infinitely many polyhedra of different homotopy types. Thus, there exist polyhedra with nilpotent fundamental groups dominating infinitely many polyhedra of different homotopy types.

On the other hand, we proved that polyhedra with finite fundamental groups and nilpotent polyhedra dominate only finitely many different homotopy types.

Applying these results and the methods used here, we also obtained solutions to some other problems concerning homotopy dominations.

Date received: February 28, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cakw-17.