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Topological K-theory: problem set 7 Problem 1. Show that for all n ≥ 3, πn (S3 ) ∼ = π n ( S2 ). Problem 2. Compute K ∗ (CPn ) as an additive group (the ring structure is more complicated). Problem 3. A prespectrum is a sequence of spaces En with maps φn : En → ΩEn+1 (the difference with a spectrum is that we don’t assume the φn to be homotopy equivalences). For a prespectrum E, define a new sequence of spaces E# by En# = hocolim Ωk En+k , where the maps in the hocolim are induced by the φn . Show that E# is a spectrum. It is called the spectrification of E. In particular, for a pointed space X, we can apply this to the sequence of spaces En = Σn X, where the structure maps φn are given by X → ΩΣX, the adjoint of the identity on ΣX. The resulting spectrum is called the suspension spectrum of X and is denoted by Σ∞ X. Problem 4. Let E be a spectrum. Show that the functor X 7→ π0 (( E ∧ X+ )#−n ) defines a homology theory. Here X+ = X t {∗} denotes the union with a disjoint base point, and E ∧ X is the prespectrum defined by ( E ∧ X )n = En ∧ X. Thus we not only get a cohomology theory out of a spectrum E, but also a homology theory.