By Joseph Neisendorfer
The main sleek and thorough remedy of volatile homotopy conception 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 variety of elements of risky homotopy idea, together with: homotopy teams with coefficients; localization and final touch; 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 publication is acceptable for a path in volatile homotopy idea, following a primary direction in homotopy conception. it's also a useful reference for either specialists and graduate scholars wishing to go into the sphere.
Read Online or Download Algebraic Methods in Unstable Homotopy Theory (New Mathematical Monographs) PDF
Similar topology books
This ebook is an creation to manifolds at first graduate point, and available to any scholar who has accomplished an exceptional undergraduate measure in arithmetic. It comprises the fundamental topological principles which are wanted for the additional examine of manifolds, rather within the context of differential geometry, algebraic topology, and comparable fields.
This e-book relies on a graduate direction taught by way of the writer on the collage of Maryland. The lecture notes were revised and augmented by way of examples. the 1st chapters increase the easy conception of Artin Braid teams, either geometrically and through homotopy idea, and talk about the hyperlink among knot idea and the combinatorics of braid teams via Markou's Theorem.
The outline for this booklet, Convergence and Uniformity in Topology. (AM-2), can be impending.
- Boundedly Controlled Topology: Foundations of Algebraic Topology and Simple Homotopy Theory (Lecture Notes in Mathematics)
- Open Problems in Topology II (Pt. 2)
- Tel Aviv Topology Conference: Rothenberg Festschrif : International Conference on Topology, June 1-5, 1998 Tel Aviv (Contemporary Mathematics)
- Differential Topology, Foliations, and Group Actions: Workshop on Topology January 6-17, 1992 Pontificia Universidade Catolica, Rio De Janeiro, Braz (Contemporary Mathematics)
- Topologie, Edition: 8. Aufl. 2005. 2., korr. Nachdruck
- Topological analysis of dynamical systems
Extra info for Algebraic Methods in Unstable Homotopy Theory (New Mathematical Monographs)
Up to p−completion, simply connected spaces with π2 torsion are locally equivalent to all their n−connected covers. In addition, Miller’s theorem  asserts that simply connected finite complexes are local in this theory with K(Z/pZ, 1) → ∗ inverted. A lemma due to Zabrodsky shows that all K(π, n) → ∗ are inverted with π a p−primary torsion abelian group. All these Eilenberg-MacLane spaces are equivalent to a point in this localization. In fact, if X is a simply connected finite complex with π2 (X) torsion and X < n > is an n−connected cover of X, then the p− completion of this kind of localization of X < n > is just the p−completion of X.
Proof: a) If X and Y are local, then there are equivalences map(M, X × Y ) ∼ = map(M, X) × map(M, Y ) Hence, X × Y is local. X × Y. 1. DROR FARJOUN-BOUSFIELD LOCALIZATION 39 b) Let A → B be a local equivalence, C be any space, and X be any local space. The space map(C, X) is local since there are equivalences map(M, map(C, X)) ∼ = map(M × C, X) ∼ = map(C, map(M, X)) map(C, X). It follows that the map A × C → B × C is a local equivalence since there are equivalences map(A×C, X) ∼ = map(A, map(C, X)) map(B, map(C, X)) ∼ = map(B×C, X).
2) the space of pointed maps map∗ (M, X) is weakly contractible. The equivalence of the above two conditions is a consequence of the fibration sequence map∗ (M, X) → map(M, X) → X. Thus, X is M − null if and only if, for all n ≥ 0, πn (map∗ (M, X)) = [Σn M, X]∗ = ∗, in other words, all pointed maps Σn M → X must be homotopic to the constant. In this sense, M looks like a point with respect to X. It is convenient that the basic definitions of localization come in two equivalent versions, pointed and unpointed.