The Toda bracket in the homotopy category of a track bicategory

✍ Scribed by K.A. Hardie; K.H. Kamps; H.J. Marcum

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 235 KB
• Volume: 175
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(02)00131-7

✦ Synopsis

We show that the tertiary homotopy composition operation (quaternary Toda bracket) can, with the advantage of reduced indeterminacy, be replaced by a triple Toda bracket in a bicategory with zeros and invertible 2-morphisms. We deÿne the bracket via a pasting operation in such a category, recovering standard properties in the new setting, and apply it to a certain bicategory whose objects are pointed maps, 1-morphisms are squares with commuting homotopies, and whose 2-morphisms are 2-tracks (Appl. Cat. Str. 8 (2000) 209). We indicate that Hopf invariant detection formulae involving the new triple bracket yield a natural extension of the classical Hopf-Toda technique for constructing non-trivial homotopy classes.