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Transactions of the American Mathematical Society

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



Probing L-S category with maps

Author: Barry Jessup
Journal: Trans. Amer. Math. Soc. 339 (1993), 351-360
MSC: Primary 55M30; Secondary 55P60, 55P62
MathSciNet review: 1112375
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Abstract: For any map $ X\xrightarrow{f}Y$, we introduce two new homotopy invariants, $ {\text{dcat}}\;f$ and $ {\text{rcat}}\;f$. The classical category $ {\text{cat}}\;f$ is a lower bound for both, while $ {\text{dcat}}\;f \leq {\text{cat}}\;X$ and $ {\text{rcat}}\;f \leq {\text{cat}}\;Y$. When $ Y$ is an Eilenberg-Mac Lane space, $ f$ represents a cohomology class and $ {\text{dcat}}\;f$ often gives a good estimate for $ {\text{cat}}\;X$. We prove that if $ \Omega \in {H^n}(M;\mathbb{Z})$ is the fundamental class of a compact, simply connected $ n$-manifold, then $ {\text{dcat}}\;\Omega = {\text{cat}}\;M$. Similarly, when $ X$ is sphere, then $ f$ is a homotopy class and while $ {\text{cat}}\;f = 1$, $ {\text{rcat}}\;f$ can be a good approximation to $ {\text{cat}}\;Y$. We show that if $ \alpha \in {\pi _2}(\mathbb{C}{P^n})$ is nonzero, then $ {\text{rcat}}\;\alpha = n$. Rational analogues are introduced and we prove that for $ u \in {H^\ast}(X;\mathbb{Q})$, $ {\text{dcat}_0}\;u = 1 \Leftrightarrow {u^2} = 0$ and $ u$ is spherical.

References [Enhancements On Off] (What's this?)

  • [Be-Ga] I. Berstein and T. Ganea, The category of a map and of a cohomology class, Fund. Math. 50 (1961/62), 265-279. MR 0139168 (25:2604)
  • [B-G] A. K. Bousfield and V. K. A. M. Gugenheim, On $ PL$ De Rham theory and rational homotopy type, Mem. Amer. Math. Soc., vol. 8, no. 179, 1976. MR 0425956 (54:13906)
  • [Fe-Ha] Y. Felix and S. Halperin, Rational L.-S. category and its applications, Trans. Amer. Math. Soc. 273 (1982), 1-37. MR 664027 (84h:55011)
  • [Fo] R. H. Fox, On the Lusternik-Schnirelmann category, Ann. of Math. (2) 42 (1941), 333-370. MR 0004108 (2:320f)
  • [G] W. J. Gilbert, Some examples for weak category and conilpotency, Illinois J. Math. 12 (1968), 421-432. MR 0231375 (37:6930)
  • [Gi] M. Ginsburg, On the L.S. category, Ann. of Math. (2) 77 (1963), 538-551. MR 0149489 (26:6976)
  • [Ha1] S. Halperin, Lectures on minimal models, Mém. Soc. Math. France (N.S.) 9-10 (1983). MR 736299 (85i:55009)
  • [Ha2] -, Finiteness in the minimal models of Sullivan, Trans. Amer. Math. Soc. 230 (1977), 173-199. MR 0461508 (57:1493)
  • [Ha-Le] S. Halperin and L.-M. Lemaire, Notions of category in differential algebra, Lecture Notes in Math., vol. 1318, Springer-Verlag, Berlin and New York, pp. 138-154. MR 952577 (89h:55023)
  • [Le] J. M. Lemairè and F. Sigrist, Sur les invariants d'homotopie rationelle lié à la L.S. catégorie, Comment. Math. Helv. 56 (1981), 103-122. MR 615618 (82g:55009)
  • [To] G. H. Toomer, Lusternik-Schnirelmann category and the Moore spectral sequence, Math. Z. 138 (1974), 175-180. MR 0356037 (50:8509)

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Keywords: Lusternik-Schnirelmann category, minimal models
Article copyright: © Copyright 1993 American Mathematical Society

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