The homotopy groups of an \(L_2\)-localized type one finite spectrum at the prime 2 (Q1392516)

This paper makes a step toward computing \(\pi_*(L_2S^0)_{(2)}\), the homotopy groups of the \(L_2\)-localization of the sphere spectrum. This step is to calculate \(\pi_*(L_2W)_{(2)}\), where \(W\) is a certain complex of type 1, which means that it satisfies \(K(0)_*(W)=0\) and \(K(1)_*(W)\neq 0\). The method is the Adams-Novikov spectral sequence. A comparison is made of these results with the Hopkins-Gross theorem relating the \(L_n\)-localization of the Spanier-Whitehead dual of a type \(n\) finite complex and the Brown-Comenetz dual, which holds for large primes.