International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 62, Pages 3903-3920
On Pierce-like idempotents and Hopf invariants
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 with cardinality , a wedge
decomposition of a space indexed by , and a cogroup ,
the homotopy group is shown, by using Pierce-like
idempotents, to have a direct sum decomposition indexed by
which is strictly functorial if is abelian.
Given a class , there is a Hopf invariant
on which extends Hopf's definition when is a comultiplication. Then is a functorial sum of over , . Each is a
functorial composition of four functors, the first depending only
on , the second only on , the third only on ,
and the fourth only on . There is a connection here with
Selick and Walker's work, and with the Hilton matrix calculus, as
described by Bokor (1991).