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 Sn. 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 → Tn into the n-torus; the same conclusion holds for X embeddable in the nth symmetric product of a curve. The existence of an algebraically essential mapping X Tn is equivalent to the existence of some elements a1; ⋯ an 2 H1(X) whose cup product a1 ⋯ an 2 Hn(X) is not zero, and implies that rankH1(X) n and catX > n. In particular, it follows that Sn, 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.