Probing L-S category with maps
Author:
Barry Jessup
Journal:
Trans. Amer. Math. Soc. 339 (1993), 351-360
MSC:
Primary 55M30; Secondary 55P60, 55P62
DOI:
https://doi.org/10.1090/S0002-9947-1993-1112375-X
MathSciNet review:
1112375
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Abstract | References | Similar Articles | Additional Information
Abstract: For any map , we introduce two new homotopy invariants,
and
. The classical category
is a lower bound for both, while
and
. When
is an Eilenberg-Mac Lane space,
represents a cohomology class and
often gives a good estimate for
. We prove that if
is the fundamental class of a compact, simply connected
-manifold, then
. Similarly, when
is sphere, then
is a homotopy class and while
,
can be a good approximation to
. We show that if
is nonzero, then
. Rational analogues are introduced and we prove that for
,
and
is spherical.
- [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
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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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1993-1112375-X
Keywords:
Lusternik-Schnirelmann category,
minimal models
Article copyright:
© Copyright 1993
American Mathematical Society