A cellular construction of BP and other irreducible spectra

Abstract

In this note we construct and derive the basic properties of the Brown-Peterson spectrum BP by attaching cells to the sphere spectrum S o (localized at a prime p) so as to kill the odd-dimensional homotopy groups. This procedure is, of course, entirely analogous to the construction of the Eilenberg-MacLane spectrum K(1g(p)) by killing the positive dimensional homotopy groups of S ~ One can thus view BP as lying half-way between S o and K(1g(p)). The advantage of our approach is that it avoids computations with Steenrod operations inherent in the original Postnikov tower construction [2]. Moreover, we obtain the homotopy and cohomology groups of BP immediately from the construction by a simple application of obstruction theory and the Adams spectral sequence. Pedagogically, we have found this approach useful in introducing BP to students who have mastered homotopy theory including the Adams spectral sequence, as for example from the texts of Moser-Tangora [5] or Switzer [9]. This paper consists of four sections the first of which gives the construction of our candidate X for the BP spectrum after making precise the notion of attaching cells non-trivially. X-~BP follows easily from the fact that a self map of a complex with cells attached non-trivially is an equivalence iff it is an equivalence on the bottom cell. In w 2 we derive the main properties of X directly from the construction without assuming the existence of BP. The idea is to prove H* (X;Z/p) is free over A/(~), then use the Adams spectral sequence to simultaneously compute rc, X and show H*(X;1g/p) is monogenic. In w we analyze the individual spaces of our spectrum. They are found to be equivalent to certain spaces in Wilson's spectrum B P @ ) . Finally in w we show that by attaching cells to kill the ( 4 k 1) dimensional homotopy groups of S o we obtain an interesting spectrum which is equivalent to MSp through the 30-skeleton.