International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 62, Pages 3903-3920

On Pierce-like idempotents and Hopf invariants

Giora Dula1 and Peter Hilton2

1Netanya College, P.O. Box 120, Neot Ganim, Netanya 42365, Israel
2Department of mathematical Sciences, State University of New York at Binghamton, 13902-6000, NY, USA

Received 3 March 2003

Copyright © 2003 Giora Dula and Peter Hilton. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Given a set K with cardinality K=n, a wedge decomposition of a space Y indexed by K, and a cogroup A, the homotopy group G=[A,Y] is shown, by using Pierce-like idempotents, to have a direct sum decomposition indexed by P(K){ϕ} which is strictly functorial if G is abelian. Given a class ρ:XY, there is a Hopf invariant HIρ on [A,Y] which extends Hopf's definition when ρ is a comultiplication. Then HI=HIρ is a functorial sum of HIL over LK, L2. Each HIL is a functorial composition of four functors, the first depending only on An+1, the second only on d, the third only on ρ, and the fourth only on Yn. There is a connection here with Selick and Walker's work, and with the Hilton matrix calculus, as described by Bokor (1991).