On contractible n-dimensional compacta, non-embeddable into R²ⁿ
by
Arkady Skopenkov
Kolmogorov College
Coauthors: Dusan Repovs

We present a very short proof of the following well-known result, which answers a question of [DaDr96] and was first proved in [RSS95; Corollary 1.5] (see also [KaRe]): For each n there exists a contractible n-dimensional compactum, non-embeddable into R²ⁿ.

We use the construction of [RSS95], modified in [KaRe], and an idea from [RSS95]. However, instead of using the Weber result of 1967 on embeddability of polyhedra in R^m, we apply its corollary, to the effect that for every n there exists a contractible n-polyhedron X, for which there is no equivariant map from the deleted product of X to S^2n-1. A simple proof of this corollary was presented by Schepin in 1984 and in [KaRe].

[DaDr96] R. J. Daverman and A. N. Dranishnikov, Cell-like maps and aspherical compacta, Illinois J. Math., 40 (1996), 77-90.

[KaRe] U. H. Karimov and D. Repovs, On embeddability of contractible k-dimensional compacta into R^2k, Topol. Appl., to appear.

[RSS95] D. Repovs, A. B. Skopenkov and E. V. Schepin, On embeddability of X×I into Euclidean space, Houston J. Math, 21 (1995), 199-204.

Date received: June 19, 2000

Copyright © 2000 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 # caeu-27.