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Transactions of the American Mathematical Society

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



Two special cases of Ganea's conjecture

Author: Jeffrey A. Strom
Journal: Trans. Amer. Math. Soc. 352 (2000), 679-688
MSC (1991): Primary 55M30, 55P50; Secondary 55P42
Published electronically: September 17, 1999
MathSciNet review: 1443893
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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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  • 1. Blakers and W. Massey: The homotopy groups of a triad, II. Ann. Math. (1952) 192-201. MR 13:485f
  • 2. T. Ganea: Some problems on numerical homotopy invariants. Lecture Notes in Mathematics 249 (1971) 23-30. MR 49:3910
  • 3. W. J. Gilbert: Some examples for weak category and conilpotency, Ill. J. Math. 12 (1968), 421-432. MR 37:6930
  • 4. K. P. Hess: A proof of Ganea's conjecture for rational spaces. Topology, 30 (1991), 205-214. MR 92d:55012
  • 5. N. Iwase: Ganea's conjecture on Lusternik-Schnirelmann category. Bull. London Math. Society 30 (1998), 623-634. CMP 98:17
  • 6. I. M. James: On category in the sense of Lusternik and Schnirelmann. Topology, 17 (1978), 331-348. MR 80i:55001
  • 7. L. Montejano: A quick proof of Singhof's $\operatorname{cat}(M\times S^1)= \operatorname{cat}(M)+1$ theorem. Manuscripta Math., 42 (1983), 49-52. MR 85a:55002
  • 8. Y. Rudyak: On category weight and its applications. Topology, 38 (1999), 37-55. MR 99f:55007
  • 9. W. Singhof: Minimal coverings of manifolds with balls. Manuscripta Math., 29 (1979), 385-415. MR 80k:55012
  • 10. P. A. Schweitzer: Secondary cohomology operations induced by the diagonal mapping. Topology. 3 (1965), 337-355. MR 32:451
  • 11. J. Strom, Category weight and essential category weight, Ph.D thesis, University of Wisconsin (1997).
  • 12. R. Switzer: Algebraic Topology: Homotopy and Homology. Springer-Verlag (1975). MR 52:6695
  • 13. G. W. Whitehead: Elements of Homotopy Theory. Springer-Verlag (1978). MR 80b:55001

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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

Received by editor(s): January 23, 1997
Published electronically: September 17, 1999
Article copyright: © Copyright 1999 American Mathematical Society