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By Joseph Neisendorfer

The main smooth and thorough remedy of volatile homotopy idea on hand. the point of interest is on these equipment from algebraic topology that are wanted within the presentation of effects, confirmed by means of Cohen, Moore, and the writer, at the exponents of homotopy teams. the writer introduces a number of facets of volatile homotopy thought, together with: homotopy teams with coefficients; localization and of entirety; the Hopf invariants of Hilton, James, and Toda; Samelson items; homotopy Bockstein spectral sequences; graded Lie algebras; differential homological algebra; and the exponent theorems about the homotopy teams of spheres and Moore areas. This e-book is acceptable for a direction in volatile homotopy concept, following a primary path in homotopy concept. it's also a worthwhile reference for either specialists and graduate scholars wishing to go into the sector.

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24 CHAPTER 1. HOMOTOPY GROUPS WITH COEFFICIENTS Finally, if p is a prime, recall that Z(p∞ ) = Z[1/p]/Z is the sequential direct limit of Z/pZ ⊂ Z/p2 Z ⊂ Z/p3 Z ⊂ . . and thus πn (X; Z(p∞ )) = lim πn (X; Z/p Z). → Exercises: 1. a) Let p∞ G = {x ∈ G : ∀r ≥ 0, ∃y ∈ G such that pr y = x} and let G = {x ∈ G : ∃r ≥ 0 such that pr x = 0} = the p−torsion subgroup of G. Show that p∞ G ⊗ Z(p∞ ) = G/p∞ G, T orZ (G, Z(p∞ )) =p∞ G. and (Z/pr Z) ⊗ Z(p∞ ) = Z/pr Z, (Z/qZ) ⊗ Z(p∞ ) = 0, T orZ (Z/pr Z, Z(p∞ )) = Z/pr Z, T orZ (Z/qZ, Z(p∞ )) = 0 if q and p are relatively prime.

8. THE MOD K HUREWICZ ISOMORPHISM THEOREM 29 Similarly, if n ≥ 3 and if π (K(A, 1); Z/kZ) = 0 for all 1 ≤ ≤ n − 1, then π (K(A, 1); Z/dZ) = 0 for all d dividing k and = 1, 2. Thus, π (K(A, 1); Z/pZ) = H (K(A, 1); Z/pZ) = 0 for all primes p dividing k and all ≥ 1. As before, induction on the number of factors of k, the strong form of the five lemma, and long exact Bockstein sequences show that π (K(A, 1); Z/kZ) = H (K(A, 1); Z/kZ) = 0 for all ≥ 1. The mod k Hurewicz theorem is true for K(A, 1) and all n ≥ 1.

DROR FARJOUN-BOUSFIELD LOCALIZATION 43 d) Show that, if F and B are both local with respect to M → ∗, then so is E. e) Give an example to show that F and E can be local without B being local. 4. Let A → B → C be a cofibration sequence and X a pointed space. a) Show that map(C, X) → map(B, X) → map(A, X) is a fibration sequence. b) Show that map∗ (C, X) → map∗ (B, X) → map∗ (A, X) is a fibration sequence. c) Suppose Y is the homotopy direct limit of a sequence Xn of spaces each of which is locally equivalent to a point with respect to M → ∗.

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