Torsion in Homotopy Groups

Abstract

Fix a prime p. A simply connected space X has exponent pr at the prime p if pr annihilates the p-primary torsion of wc*(X). The basic example showing that such spaces exist is the fact that S2I'+' has exponent p2f` at p. This result was first obtained in the case p equals 2 by I. M. James [17] and later in the case of an odd prime by H. Toda [41]. We wished to find further information concerning exponents. Indeed we started by wondering whether Moore spaces of finite abelian groups have an exponent. While we do not answer this question, we do obtain further information concerning the structure of their torsion. In particular, for Moore spaces with nontrivial homology group Z/prZ where p is greater than three, we exhibit an infinite family of elements of order pr+l. These are obtained in making product decompositions of certain associated loop spaces. These decompositions are constructed by using Samelson products in mod p homotopy theory. They lead to information concerning the double suspension. This enables us to improve the result of Toda by showing that S2'f+' has exponent p'+1 at p. In a subsequent paper, we will show that the exponent of S2'f+' at p is in fact pfl by refining our results on the double suspension and using P. Selick's theorem [34] that S3 has exponent p at any odd prime p. Our methods, which involve differential Lie algebras, are both elementary and accessible. Higher torsion is detected by Samelson products and the homology of free differential Lie algebras. Product decompositions of loop spaces are constructed from analogous decompositions of universal enveloping algebras. The exponent of S2'f+l is a consequence of a map Q2 -~ffl S'(P)-1 derived from one such product decomposition. The considerations in this paper lead us to wonder whether simply connected finite dimensional H-spaces have exponents at all primes or, more