By M.A. Armstrong
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Extra info for Basic topology
The restriction of f, to the bottom cell is 31. The fact that this extends over the third cell means that a1Qi = 0 in This means that h1,Obi,o must be the target of a differential in the Adams spectral sequence for the sphere spectrum. If one computes the Adams E2-term through the relevant range of dimensions, one finds that the only possible source for this differential is b1,1. However, one can also deduce this by studying the map fP more closely. Let So= Y0(Y1(Y2( ... be an Adams resolution for S°.
We now turn to the most basic of these nonconnective periodic localizations, namely, the K*-localization. This is the same as the KO. -localization since K. and KO* (as well as K* and KO*) have the same equivalences in S. We believe that the K*-localization was first studied somewhat speculatively by Adams-Baird and then more substantively by Ravenel. 2 Theorem. For a spectrum X, there is a natural equivalence XK SK A X where SK is the "nonconnective ImJ spectrum" determined at a prime p by cofiber sequences KO(p) (S -)(p) J(p) -r E-1 HQ with r = 3 for p = 2 and with r an integer generating the group of units in Z/p2 for p odd.
The calculations we give here follow a different, and more succinct, approach to that of our original ones in . ) Hopf ring in joint work with M. J. Hopkins. This will compute the Hopf ring directly from the Landweber exactness property of E(n)-theory. Hunton : Detruncating Morava K-theory 37 All singular homology groups used will be with mod p coefficients (p odd) and from now on we shall suppress this from the notation. All homology is unreduced. We assume a certain familiarity with the basic notions and terminology of Hopf rings; the reader will find  a suitable general reference.
Basic topology by M.A. Armstrong