Algebraic Methods in Unstable Homotopy Theory (New by Joseph Neisendorfer

By Joseph Neisendorfer

The main sleek 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 through Cohen, Moore, and the writer, at the exponents of homotopy teams. the writer introduces a number of elements of risky homotopy idea, 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 publication is appropriate for a path in risky homotopy concept, following a primary path in homotopy thought. it's also a useful reference for either specialists and graduate scholars wishing to go into the sector.

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The Hurewicz theorem implies that ∆ is null homotopic. Hence [f, g] is null homotopic and π2 (ΩX; Z/kZ) is abelian if k is odd. 34 CHAPTER 1. K. Bousfield [15, 16]. The theory is founded on the homotopy theoretic consequences of inverting a specific map µ of spaces. Those spaces for which the mapping space dual of µ is an equivalence are called local. In turn, the local spaces define a set of maps called local equivalences. The localization of a space X is defined to be a universal local space which is locally equivalent to X.

First applications occur in the theory of H-spaces and are due to Sullivan, to Peter Hilton and Joseph Roitberg [53], and to Alexander Zabrodsky [142] . There are two themes in localization theory. One is to study a space in more depth by inverting some or all primes. For example, Serre’s result [118] that ΩS 2n+2 S 2n+1 × ΩS 4n+3 is valid once 2 is inverted but not before unless n = 0, 1, 3. [2] The extreme example of this theme is rationalization, inverting all 35 36 CHAPTER 2. A GENERAL THEORY OF LOCALIZATION primes.

Show that G ⊗ Z[1/p] = 0 if G = Z/pr Z Z/qZ if G = Z/qZ with q and p relatively prime. 3. Let X be a simply connected CW complex with πn (X; Z[1/p]) = 0, πn (X; Z(p∞ )) = 0 for all n ≥ 2. Show that X is contractible. 4. Suppose X is a simply connected space. Show that πn (X; Q/Z) = 0 for all n ≥ 2 if and only if πn (X) is a rational vector space for all n ≥ 2. 5. Suppose X is a simply connected space. Show that πn (X; Q) = 0 for all n ≥ 2 if and only if πn (X) is a torsion group for all n ≥ 2. 7.

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