Embeddings into products and symmetric products - An algebraic approach

Akira Koyama, Józef Krasinkiewicz, Stanis Law Spiez

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2 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)611-641
Number of pages31
JournalHouston Journal of Mathematics
Volume38
Issue number2
Publication statusPublished - 2012 Dec 1

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Keywords

  • Cohomology groups
  • Embeddings
  • Products
  • Symmetric products

ASJC Scopus subject areas

  • Mathematics(all)

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