By Joseph Neisendorfer

ISBN-10: 0521760372

ISBN-13: 9780521760379

The main sleek and thorough remedy of risky homotopy idea to be had. 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 points of risky homotopy thought, 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 about the homotopy teams of spheres and Moore areas. This e-book is appropriate for a direction in volatile homotopy concept, following a primary direction in homotopy conception. it's also a worthy reference for either specialists and graduate scholars wishing to go into the sector.

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**Additional info for Algebraic Methods in Unstable Homotopy Theory**

**Example text**

A) Suppose there is a positive integer r such that ps πm (X; Z/ps Z) = 0 for all s ≤ r. If αεTorZ (πm −1 (X), Z/pr z) has order ps with s ≤ r, then there is an element γεπm (X; Z/pr Z) which has order ps and such that γ maps to α in the universal coefficien sequence 0 → πm (X) ⊗ Z/pr Z → πm (X; Z/pr Z) → TorZ (πm −1 (X), Z/pr Z) → 0. (b) If πm −1 (X) is finitel generated,together with the hypotheses in (a), show that the above universal coefficien sequence for Z/pr Z coefficient is split. 1. If f : G → H is a homomorphism of finitel generated abelian groups and n ≥ 2, then there exists a map F : P n (H) → P n (G) such that the induced cohomology map F ∗ = f.

It follows that there are unnatural isomorphisms F ∗ ∼ = F and T ∗ ∼ = T. The following lemma is easily verifie in the cyclic case and hence in all cases. 2. For finitel generated free F and finit T , the natural maps F → (F ∗ )∗ and T → (T ∗ )∗ are isomorphisms. 3. For finitel generated generated free F1 and F2 and finit T1 and T2 , the natural maps Hom(F1 , F2 ) → Hom(F2∗ , F1∗ ) and Hom(T1 , T2 ) → Hom(T2∗ , T1∗ ) sending a homomorphism f to its dual f ∗ are isomorphisms. 4. For finit abelian T , there is a natural isomorphism T ∗ ∼ = Ext(T, Z).

83in 978 0 521 76037 9 December 26, 2009 Homotopy groups with coefficients If X is simply connected the last term is 0. Otherwise, consider the twist map T : P 2 (Z/kZ) × P 2 (Z/kZ) → P 2 (Z/kZ) × P 2 (Z/kZ), T (x, y) = (y, x). Since T ◦ ∆ = ∆, it follows that λ = −λ, or 2λ = 0. If k is odd, then λ = 0. Remark. Since P 2 (Z/2Z) is just the two-dimensional projective space, the well known computation of the mod 2 cup product shows that λ = 1 and ϕ([f ] + [g]) = ϕ([f ]) + ϕ([g]) + f∗ (s1 ) · g∗ (s1 ) in the case: ϕ : π2 (X; Z/2Z) → H2 (X; Z/2Z) with X a homotopy associative H-space.

### Algebraic Methods in Unstable Homotopy Theory by Joseph Neisendorfer

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