Novikov’s Ext 2 and the nontriviality of the gamma family

Larry Smith [7] defined and detected elements j3f in the p-primary compo-nent of the stable homotopy of the sphere for t> 0 and p> 5. In the same manner, Toda's construction [11] gives elements yt f or t> 0 and p>l. We have the following results which are a consequence of our computation of the second line of the Novikov spectral sequence for a sphere at an odd prime. THEOREM 1. (a) p does not divide 0, G 7 r ^ 2 _ i) f 2 ( p _ i) _ 2 ( 5, ° ) forp>5, t>0. (b) 0 * 7, 6 i r ^ s., , ^ ^ , ^ ^ ^ ) forp>l,t> 0. (c) axpt =£0fort=£0or-l mod p,p>5. Partial results on the nontriviality of yt have been obtained by Thomas and Zahler [10], [9], Oka and Toda [6], Johnson, Miller, Wilson, and Zahler [2], and Ravenel (unpublished). These infinite families can be studied most conveniently by means of the Novikov spectral sequence E* * = Ext**>sp(5P*, BP*(X)) = * 7r*(X)(p) for a space X [1]. BP* ( ) is the Brown-Peterson homology theory [1], and BP * = BP*(S°) = Z(p)[vl9v2,...], \vt \ = 2(pf- 1). Let In denote the invariant ideal (pf vt,..., vn_l) C BP* \ I0 = (0). For a BP*BP comodule M let H*M denote Ext^p BP(BP*9 M). By a theorem of Land-web er [3] we have for n> 0 H°BPJI „ = Fp[vn]. Let 8n: HlBPjIn+l —+ Hi+lBP#/In be the connecting homomorphism in the long exact sequence associated with 0- + BPjIn — % B P j I n-+ BPjIn+l- * 0. It is folklore (see [2]) that if p> 7, t> 0, and 0 * ô0ô1Ô2(u|) E H3BP*, then this class survives to yt and yt ¥ = 0. Our proof of Theorem 1 involves an analysi