Generalized Eilenberg Theorem and its Applications
by
A. N. Dranishnikov
University of Florida

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 \tauK 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 \tauSⁿ 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 \tauK * 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 \tauL.

Date received: January 31, 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 Conferences Inc. Document # caaa-44.