By Joseph Neisendorfer

ISBN-10: 0521760372

ISBN-13: 9780521760379

The main glossy and thorough therapy of volatile homotopy thought on hand. the point of interest is on these equipment from algebraic topology that are wanted within the presentation of effects, confirmed through Cohen, Moore, and the writer, at the exponents of homotopy teams. the writer introduces a number of features of volatile homotopy concept, together with: homotopy teams with coefficients; localization and crowning glory; the Hopf invariants of Hilton, James, and Toda; Samelson items; homotopy Bockstein spectral sequences; graded Lie algebras; differential homological algebra; and the exponent theorems in regards to the homotopy teams of spheres and Moore areas. This e-book is appropriate for a direction in risky homotopy thought, following a primary path in homotopy thought. it's also a useful reference for either specialists and graduate scholars wishing to go into the sphere.

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**Example text**

Suppose we have a pointed map g : Σk (M ) → LM (A). Since each Σk (Mn ) is a finite complex, its image is contained in some Lαn (A) for an ordinal αn εΩ. Thus the image of Σk (M ) is contained in the countable limit Lγ (A) with γ = sup αn εΩ. Thus, g is null homotopic in the mapping cone Lγ+1 (A). Since Lγ+1 (A) ⊂ LM (A), g is null homotopic in LM (A). Hence, LM (A) is local. Second, we show that A → LM (A) is a local equivalence. For all local X and k ≥ 0, we need bijections [Σk (LM (A)), X]∗ → [Σk (A), X]∗ .

5. THE BOCKSTEIN LONG EXACT SEQUENCE 21 is a pushout diagram and there is a cofibration sequence X1 ∪A Y1 → Z1 → Z1 /X1 ∪A Y1 . Proof: First replace A → X and A → Y by cofibrations A → X1 and A → Y1 . Then use the homotopy extension property of the cofibration A → X1 to make the diagram strictly commutative. The inclusions X1 → X1 ∪A Y1 and Y1 → X1 ∪A Y1 are cofibrations. Replace the map X1 ∪A Y1 → Z by a cofibration X1 ∪A Y1 → Z1 . The rest follows by collapsing subspaces. For example, the homotopy commutative diagram S n−1 ↓k − → 1 S n−1 ↓k S n−1 → − S n−1 yields the homotopy commutative diagram below in which all rows and columns are cofibration sequences S n−1 ↓k − → 1 S n−1 ↓k → ∗ ↓ → Sn ↓k → S n−1 ↓ → − S n−1 ↓ → P n (Z/ Z) ↓1 → Sn ↓ → P n (Z/kZ) − → ρ η β P n (Z/k Z) − → P n (Z/ Z) − → P n+1 (Z/kZ) → .

On the other hand, if π1 (K(A, 1), Z/pZ) = Ap = 0, then the mod p Hurewicz map ϕ is an isomorphism in dimensions 1 and 2 and an epimorphism in dimension 3. Hence, the mod p Hurewicz theorem is true for K(A, 1) and n = 2. If π1 (K(A, 1), Z/pZ) = π2 (K(A, 1), Z/pZ) = 0, then πk (K(A, 1), Z/pZ) = Hk (K(A, 1), Z/pZ) = 0 for all k ≥ 1. We conclude that ϕ is an isomorphism in all dimensions. The mod p Hurewicz theorem is true for K(A, 1) and all n ≥ 1. 5: If p is a prime and n ≥ 3, then Ap A p H (K(A, m); Z/pZ) = 0 0 if if if if Ap = m, = m + 1, = m + 2 and p is odd.

### Algebraic Methods in Unstable Homotopy Theory (New Mathematical Monographs) by Joseph Neisendorfer

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