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November 2004 The Cone Length and Category of Maps: Pushouts, Products and Fibrations
Martin Arkowitz, Donald Stanley, Jeffrey Strom
Bull. Belg. Math. Soc. Simon Stevin 11(4): 517-545 (November 2004). DOI: 10.36045/bbms/1102689120

Abstract

For any collection of spaces ${\cal A}$, we investigate two non-negative integer homotopy invariants of maps: $L_{\cal A}(f)$, the ${\cal A}$-cone length of $f$, and ${\cal L}_{\cal A}(f)$, the ${\cal A}$-category of $f$. When ${\cal A}$ is the collection of all spaces, these are the cone length and category of $f$, respectively, both of which have been studied previously. The following results are obtained: (1) For a map of one homotopy pushout diagram into another, we derive an upper bound for $L_{\cal A}$ and ${\cal L}_{\cal A}$ of the induced map of homotopy pushouts in terms of $L_{\cal A}$ and ${\cal L}_{\cal A}$ of the other maps. This has many applications, including an inequality for $L_{\cal A}$ and ${\cal L}_{\cal A}$ of the maps in a mapping of one mapping cone sequence into another. (2) We establish an upper bound for $L_{\cal A}$ and ${\cal L}_{\cal A}$ of the product of two maps in terms of $L_{\cal A}$ and ${\cal L}_{\cal A}$ of the given maps and the ${\cal A}$-cone length of their domains. (3) We study our invariants in a pullback square and obtain as a consequence an upper bound for the ${\cal A}$-cone length and ${\cal A}$-category of the total space of a fibration in terms of the ${\cal A}$-cone length and ${\cal A}$-category of the base and fiber. We conclude with several remarks, examples and open questions.

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Martin Arkowitz. Donald Stanley. Jeffrey Strom. "The Cone Length and Category of Maps: Pushouts, Products and Fibrations." Bull. Belg. Math. Soc. Simon Stevin 11 (4) 517 - 545, November 2004. https://doi.org/10.36045/bbms/1102689120

Information

Published: November 2004
First available in Project Euclid: 10 December 2004

zbMATH: 1078.55006
MathSciNet: MR2115724
Digital Object Identifier: 10.36045/bbms/1102689120

Subjects:
Primary: 55M30
Secondary: 55P99, 55R05

Rights: Copyright © 2004 The Belgian Mathematical Society

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Vol.11 • No. 4 • November 2004
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