The cone length and category of maps: pushouts, products and fibrations

Original · EN

For any collection of spaces A, we investigate two non-negative integer homotopy invariants of maps: lₐ(f), the A-cone length of f, and Lₐ(f), the A-category of f. When 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 have been obtained: (1) For a map of one homotopy pushout diagram into another, we derive an upper bound for Iₐ and Lₐ of the induced map of homotopy pushouts in terms of Iₐ and Lₐ of the other maps. This has many applications including an inequality for Iₐ and Lₐ of the maps in a mapping of one mapping cone sequence into another. (2) We establish an upper bound for Iₐ and Lₐ of the product of two maps in terms of Iₐ and Lₐ of the given maps and the 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 A-cone length and A-category of the total space of a fibration in terms of the A-cone length and A-category of the base and fiber. We conclude with several remarks, examples and open questions.