0.1. CONVENTIONS xi 0.1. Conventions As stated at the beginning of the introduction, all spaces and spectra are im- plicitly localized at the prime 2. Homology and cohomology is implicitly taken with F2-coeﬃcients. Mod 2 binomial coeﬃcients a b ) ∈ F2 are defined for all a, b ∈ Z by a b = coeﬃcient of tb in (1 + t)a. We will denote the connectivity of a space Y by conn(Y ). For a spectrum E, we denote its Spanier-Whitehead dual by E∨. We shall use Sn to denote the n-sphere as a space, and use Sn to denote the n-sphere as a suspension spectrum. We will use S to denote the sphere spectrum S0. Throughout this book we will be dealing with completely unadmissible se- quences of integers (which we shall sometimes refer to briefly as “CU sequences”). A CU sequence is a sequence of integers (j1, . . . , jk) with js ≥ 2js+1 +1. We shall often refer to a CU sequence by a single capital letter (e.g. J = (j1, . . . , jk)). We associate to CU sequences the following quantities: • Length: |J| := k, • Degree: J := j1 + · · · + jk, • Excess: e(J) := jk. The empty sequence ∅ is regarded as a sequence of length zero, degree zero, and excess ∞. We introduce a total ordering on CU sequences of all lengths: we declare that (j1, . . . , jk) ≤ (j 1 , . . . , j k ) if either k k or, for k = k , if J ≤ J with respect to right lexicographical ordering. For CU sequences J = (j1, . . . , jk) J = (j1, . . . , jk ) with jk ≥ 2j 1 + 1, we shall let [J, J ] denote the CU sequence (j1, . . . , jk, j 1 , . . . , j k ).

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