Composition methods in the homotopy groups of ring spectra

Abstract

Progress in calculating the homotopy groups of spheres has seen two major breakthroughs. The first was Toda’s work, culminating in his book [11] in which the EHP sequences of James and Whitehead were used inductively; “composition methods” were used to construct elements and evaluate homomorphisms. The second was the Adams spectral sequence. Each method has advantages and disadvantages. Toda’s method has the advantage that unstable homotopy groups are calculated along with stable groups. It has the disadvantage that it applies only to spheres and in particular, naturality under maps between spaces cannot be applied. The Adams method has the advantage that much of the bookkeeping work is accomplished in advance during the calculation of the Ext groups. The disadvantage that it does not calculate unstable groups has been eliminated, for certain nice spaces, by the work of [3]. This work implies that for certain spaces, the unstable homotopy is as accessible as the stable homotopy. It is the purpose of this work to examine how, in certain cases, the methods of Toda can be used for spaces other than spheres. We will begin by summarizing the methods used by Toda. We will discuss the possibility of using these methods for other spectra and work out the example of the Moore space spectrum S0UP r e 1 for p > 3.