Essential Maps Exist From BU to coker J

Abstract

We show that [BU, coker J] :$ 0 but that there are no infinite loop maps from BU to coker J. The proofs involve the Segal conjecture. It is known that [coker J, BU] = 0 [HS]. A conjecture dating from the early 70's is that [BU, coker J] = 0 also. This conjecture is attributed by some people to Sullivan. It seems quite reasonable since BU and coker J should have little to do with each other. The purpose of this note is to show that despite this intuition, [BU, coker J] $& 0. Indeed it is uncountable. However, there are no infinite loop maps from BU to coker J. So the conjecture is true on the level of infinite loop maps. Both these statements follow from the Segal conjecture, in particular from the compact Lie group version of the Segal conjecture proven by this author. We give a proof which does not use the main theorem in [F], the compact Lie group version; the full power of this generalization is not needed. Although this result is interesting in its own right, what is particularly striking is how inaccessible such a calculation seemed just a few years ago. The author wishes to thank Ib Madsen for bringing this conjecture to his attention and for his encouragement. Conversations with S. Priddy were also helpful. LEMMA 1. [BU, QoSO] is torsion free. Hence [BU, Im J] is torsion free. PROOF. The second statement follows from the first, since QoS? Im J x coker J [MM, p. 110]. We show that 7r?(BU) is detected on finite subgroups; the result follows by the Segal conjecture. [BU, QoSO] = 7r0BU, the reduced stable cohomotopy in dimension 0 of BU. Since BU equals limBU(n), there is a Milnor short exact sequence 0 -O lim' 7ri BU(n) -i7rBU -lim7r?BU(n) -1. 7r8 'BU(n) is compact for all n by the Atiyah-Hirzebruch spectral sequence for finite skeleta. Hence, 7r?BU lim 7r?BU(n). 7r?BU(n) is detected on finite subgroups of U(n) by Corollary 2.14 [F]. Since the stable cohomotopy in dimension 0 of a finite group is torsion free by the Segal conjecture for finite groups [C], it follows that 7r?BU is torsion free. LEMMA 2. [BU, Im J] = 0. Hence [BU, coker J] = [BU, QoS?]. Received by the editors February 28, 1985. 1980 Mathematics Subject Classification. Primary 55N20; Secondary 55P42, 55Q55.