Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1993-1112375-X
MathSciNet review: 1112375
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55M30, 55P60, 55P62

Retrieve articles in all journals with MSC: 55M30, 55P60, 55P62


Additional Information

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