“Chromatic ” homotopy theory
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چکیده
Homotopy theory deals with spaces of large but finite dimension. Chromatic homotopy theory is an organizing principle which is highly developed in the stable situation. 1. The Spanier-Whitehead category. We'll work with the category of finite polyhedra (or finite CW complexes) and homotopy classes of continuous maps between them. We will always fix a basepoint in all spaces, and assume that maps and homotopies preserve them. Write F for this homotopy category, and write [K, L] for the set of pointed homotopy classes of pointed maps. It's just a pointed set, and very difficult to compute in even quite simple cases. The idea of stable homotopy theory is to try to simplify this problem by a certain type of localization. There is a shift operator on F. Embed K into the cone on K, C(K) = [0, 1] × K 1 × K ∪ [0, 1] × * as the subspace 0 × K, and then collapse this subspace to a point. This is the suspension ΣK of K. The construction may be iterated. There is a natural isomorphism H q+n (Σ n K) = H q (K) By functoriality, there are maps and these maps are eventually isomorphisms. Elements of the direct limit are called stable maps from K to L. Maps from a suspension form a group, maps from a double suspension form an abelian group, and the suspension maps are homomorphisms when they can be: so the set of stable maps forms an abelian group. Spanier and Whitehead (around 1955) defined what is now called the homotopy category of finite spectra, S.
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