caa a22 4500 44537876X CHVBK 20180317143011.0 cr unu---uuuuu 170323e20110201xx s 000 0 eng 10.1007/s10485-009-9186-3 doi (NATIONALLICENCE)springer-10.1007/s10485-009-9186-3 Long Box Bracket Operations in Homotopy Theory [Elektronische Daten] [Howard Marcum, Nobuyuki Oda] Secondary homotopy operations called box bracket operations were defined in the homotopy theory of an arbitrary 2-category with zeros by Hardie, Marcum and Oda (Rend Ist Mat Univ Trieste, 33:19-70 2001). For the topological 2-category of based spaces, based maps and based track classes of based homotopies, the classical Toda bracket is a particular example of a box bracket operation and subsequent development of the theory has refined, clarified and placed in this more general context many of the properties of classical Toda brackets. In this paper, and for the topological case only, we use an inductive definition to extend the theory to long box brackets. As is well-known, the necessity to manage higher homotopy coherence is a complicating factor in the consideration of such higher order operations. The key to our construction is the definition of an appropriate triple box bracket operation and consequently we focus primarily on the properties of the triple box bracket. We exhibit and exploit the relationship of the classical quaternary Toda bracket to the triple box bracket. As our main results we establish some computational techniques for triple box brackets that are based on composition methods. Some specific computations from the homotopy groups of spheres are included. Springer Science+Business Media B.V., 2009 2-Category nationallicence Suspension functor nationallicence Toda bracket (quaternary, long) nationallicence Box bracket (triple, long) nationallicence Homotopy groups of spheres nationallicence Marcum Howard The Ohio State University at Newark, 1179 University Drive, 43055, Newark, Ohio, USA aut Oda Nobuyuki Department of Applied Mathematics, Faculty of Science, Fukuoka University, 814-0180, Fukuoka, Japan aut Applied Categorical Structures Springer Netherlands 19/1(2011-02-01), 137-173 0927-2852 19:1<137 2011 19 10485 https://doi.org/10.1007/s10485-009-9186-3 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10485-009-9186-3 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Marcum Howard The Ohio State University at Newark, 1179 University Drive, 43055, Newark, Ohio, USA aut NATIONALLICENCE 700 1- Oda Nobuyuki Department of Applied Mathematics, Faculty of Science, Fukuoka University, 814-0180, Fukuoka, Japan aut NATIONALLICENCE 773 0- Applied Categorical Structures Springer Netherlands 19/1(2011-02-01), 137-173 0927-2852 19:1<137 2011 19 10485 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer