On the action of β1 in the stable homotopy of spheres at the prime 3
نویسنده
چکیده
The element β1 is the generator of the stable homotopy group π10(S ). Here S denotes the 3-localized sphere spectrum. Toda showed that β 1 6= 0 and β 1 = 0. Here we generalize it to β 1β9t+1 6= 0 and β 1β9t+1 = 0 for β9t+1 ∈ π144t+10(S) with t ≥ 0. In particular, β 1β10 6= 0 and β 1β10 = 0 for β10 shown to exist by Oka. This is proved by determining subgroups of π∗(L2S ), where L2 denotes the Bousfield localization functor with respect to v−1 2 BP .
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