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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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Probing L-S category with maps
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by Barry Jessup PDF
Trans. Amer. Math. Soc. 339 (1993), 351-360 Request permission

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.
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • 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