On the non-triviality of the Greek letter elements in the Adams-Novikov \(E_ 2\)-term (Q1813595)

The search for regular behaviour in the \(p\)-localised stable homotopy groups of spheres usually begins at the \(E_ 2\) term, \(H^*(\text{BP}_ *)\), of the Adams-Novikov spectral sequence. For instance it is not hard to see that \((v_ n)^ t\) gives a class in \(H^ 0( \text{BP}_ */ I_ n)\) for any \(t \geq 1\) where \(I_ n=(p,v_ 1,\ldots,v_{n-1})\). Since the sequence \(p\), \(v_ 1,\ldots,v_{n-1}\) is regular we may apply Bokshteĭns to this and obtain the so-called \(n\)th Greek letter family \(\alpha_ t^{(n)}\) in \(H^ n(\text{BP}_ *)\). We then ask if these elements are nonzero and if they survive the spectral sequence to give stable homotopy elements. The algebraic situation is quite well understood for \(n \leq 3\) [\textit{H. R. Miller}, \textit{D. C. Ravenel} and \textit{W. S. Wilson}, Ann. Math., II. Ser. 106, No. 3, 469-516 (1977; Zbl 0374.55022)] and in this range the geometric consequences flow directly from earlier constructions of Adams, Smith and Toda. The main theorem of the present paper is that for \(p \geq n \geq 3\) and \(p-1 \geq t \geq 1\) the elements \(\alpha_ t^{(n)}\) are nonzero and not divisible by \(p\). The author shows that these elements have nonzero image in \(H^*\bigl( \text{BP}_ */I_{n-1}+I_ n^{p-n+1} \bigr)\), which amounts to a study of the appropriate element in the \(v_{n-1}\)- Bokshteĭn spectral sequence for \(H^*(\text{BP}_ */I_{n-1})\). For this he constructs suitable subquotients of cobar complexes where he can calculate explicitly.