The Cone Length and Category of Maps: Pushouts, Products and Fibrations

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.