The main results:(1) Let L be a nilpotent CW complex and F the homotopy fiber of the inclusion i of L into its infinite symmetric product SP(L). If X is a metrizable space such that X\tau K(H_k(L),k) for all k\ge 1, then X\tau K(\pi_k(F),k) and X\tau K(\pi_k(L),k) for all k\ge 2. (2) Let X be a metrizable space such that \dim (X) < \infty or X\in ANR. Suppose L is a nilpotent CW complex. If X\tau SP(L), then X\tau L in the following cases: (a) H_1(L) is finitely generated, or (b) H_1(L) is a torsion group.

COBISS.SI-ID: 14551385

Two important questions are answered in the negative: (1) If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic? (2) Can adding arcs to a space cause an essential curve to become nulhomotopic? The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and \pi_1-shape injective.

COBISS.SI-ID: 14657625