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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Two special cases of Ganea’s conjecture
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by Jeffrey A. Strom PDF
Trans. Amer. Math. Soc. 352 (2000), 679-688 Request permission

Abstract:

Ganea conjectured that for any finite CW complex $X$ and any $k>0$, $\operatorname {cat}(X\times S^k) =\operatorname {cat}(X) + 1$. In this paper we prove two special cases of this conjecture. The main result is the following. Let $X$ be a $(p-1)$-connected $n$-dimensional CW complex (not necessarily finite). We show that if $\operatorname {cat}(X) = \left \lfloor {n \over p} \right \rfloor + 1$ and $n\not \equiv -1 \operatorname {mod} p$ (which implies $p>1$), then $\operatorname {cat}(X\times S^k) =\operatorname {cat}(X) +1$. This is proved by showing that $\operatorname {wcat}(X\times S^k) =\operatorname {wcat}(X) + 1$ in a much larger range, and then showing that under the conditions imposed, $\operatorname {cat}(X)=\operatorname {wcat}(X)$. The second special case is an extension of Singhof’s earlier result for manifolds.
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Additional Information
  • Jeffrey A. Strom
  • Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706
  • Address at time of publication: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
  • Email: jeffrey.strom@dartmouth.edu
  • Received by editor(s): January 23, 1997
  • Published electronically: September 17, 1999
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 352 (2000), 679-688
  • MSC (1991): Primary 55M30, 55P50; Secondary 55P42
  • DOI: https://doi.org/10.1090/S0002-9947-99-02046-2
  • MathSciNet review: 1443893