Free Mappings and Solenoidal Continua
by
Zbigniew Karno
Warsaw University at Białystok

In a recent paper J. Krasinkiewicz has proved that arbitrary manifold M with dimM >= 1 is free, as well as the product of this space with any compactum. We prove that all k-dimensional weakly solenoidal continua with manifoldal expansion are free for k >= 1, and, in particular, all solenoids are free. Actually, we prove more general property of such continua.

We call a metric space Y to be free if every mapping f : X --> Y from a compactum X into Y can be approximated by mappings with hereditarily indecomposable fibers. We call a mapping p : Y --> Z to be strongly free if every mapping f : X --> Y from any compactum X into Y can be approximated by mappings with the property that the compositions of these with p have hereditarily indecomposable fibers. Note that if Z is free and p is strongly free, then Y must also be free.

By a weakly solenoidal continuum we mean the inverse limit M of an inverse sequence (M_n, \piⁿ_m) (called an expansion of M), whose each space M_n is a compact connected manifold (with or without boundary), and each bonding mapping \piⁿ_m : M_n --> M_m (n >= m) is a covering map. Every projection \pi_n : M --> M_n is called a weakly solenoidal projection.

Our main result is the following:

Theorem Let M be a k-dimensional weakly solenoidal continuum with a manifoldal expansion, where k >= 1. Then every weakly solenoidal projection \pi_n : M --> M_n is strongly free. In particular, M is free.

Date received: June 24, 1996

Copyright © 1996 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 Mathematical Conference Abstracts. Document # caah-50.