FUNDAMENTA MATHEMATICAE 178 (2003) Homotopy dominations within polyhedra by Danuta Kolodziejczyk (Warszawa) Abstract. We show the existence of a finite polyhedron P dominating infinitely many different homotopy types of finite polyhedra and such that there is a bound on the lengths of all strictly descending sequences of homotopy types dominated by P . This answers a question of K. Borsuk (1979) dealing with shape-theoretic notions of “capacity” and “depth” of compact metric spaces. Moreover, π 1 (P ) may be any given non-abelian poly-Z-group and dim P may be any given integer n ≥ 3. 1. Introduction. In the paper, every polyhedron is assumed to be finite and connected. Following K. Borsuk, define the capacity C (A) of a compactum A as the cardinality of the class of shapes of all compacta X such that Sh(X ) ≤ Sh(A). (The basic notions and facts of shape theory can be found in [B2], [DS], [MS].) A system consisting of k compacta X 1 ,...,X k is said to be a chain of length k for a compactum A if Sh(X 1 ) <...< Sh(X k ) ≤ Sh(A) (Sh(X ) < Sh(Y ) if and only if Sh(X ) ≤ Sh(Y ) holds but Sh(Y ) ≤ Sh(X ) fails). The depth D(A) of a compactum A is the least upper bound of the lengths of all chains for A. If this upper bound is infinite, we write D(A)= ℵ 0 . Let us remark that on ANRs shape theory coincides with homotopy the- ory. Moreover, in the case where A is a polyhedron, one may replace the above definitions by their “homotopy versions” (i.e. in the homotopy ca- tegory of CW -complexes). Indeed, by the classical results in shape theory (see [HaHe1] or [HaHe2]; [DS, Theorem 2.2.6]; [EG]) there is a 1-1 functo- rial correspondence between the shapes of compacta shape dominated by a given polyhedron and the homotopy types of CW -complexes (not necessar- 2000 Mathematics Subject Classification : 55P55, 55P15. Key words and phrases : polyhedron, CW -complex, homotopy domination, homotopy type, compactum, shape domination, shape, capacity, depth. The author would like to thank the Institute of Mathematics of the Polish Academy of Sciences for its support while part of this work was done. [189]