EMBEDDINGS INTO PRODUCTS AND SYMMETRIC PRODUCTS - AN ALGEBRAIC APPROACH

In this article we mostly study algebraic properties of n-dimensional "cyclic" compacta lying either in products of n curves or in the nth symmetric product of a curve. The basic results have been obtained for compacta admitting essential maps into the n-sphere S-n. One of the main results asserts that if a compactum X admits such a mapping and X embeds in a product of n curves then there exists an algebraically essential map X -> T-n into the n-torus; the same conclusion holds for X embeddable in the nth symmetric product of a curve. The existence of analgebraically essential mapping X -> T-n is equivalent to the existence of some elements a(1), . . . ,a(n) is an element of H-1 (X) whose cup product a(1)boolean OR . . . boolean OR a(n) is an element of H-n (X) is not zero, and implies that rank H-1(X) >= n and cat X > n. In particular, it follows that S-n, n >= 2, is not embeddable in the nth symmetric product of any curve. The case n = 2 answers in the negative a question of Illanes and Nadler.