By Mark Hovey
This publication supplies an axiomatic presentation of reliable homotopy idea. It starts off with axioms defining a "stable homotopy category"; utilizing those axioms, you can still make a variety of constructions---cellular towers, Bousfield localization, and Brown representability, to call a couple of. a lot of the ebook is dedicated to those buildings and to the examine of the worldwide constitution of reliable homotopy different types.
Next, a few examples of such different types are awarded. a few of those come up in topology (the usual good homotopy type of spectra, different types of equivariant spectra, and Bousfield localizations of these), and others in algebra (coming from the illustration idea of teams or of Lie algebras, as good because the derived classification of a commutative ring). therefore one can follow a number of the instruments of good homotopy thought to those algebraic events.
Features:
Provides a reference for traditional effects and structures in solid homotopy thought.
Discusses purposes of these effects to algebraic settings, corresponding to crew idea and commutative algebra.
Provides a unified therapy of a number of various occasions in solid homotopy, together with equivariant good homotopy and localizations of the reliable homotopy classification.
Provides a context for nilpotence and thick subcategory theorems, reminiscent of the nilpotence theorem of Devinatz-Hopkins-Smith and the thick subcategory theorem of Hopkins-Smith in good homotopy thought, and the thick subcategory theorem of Benson-Carlson-Rickard in illustration concept.
This e-book provides sturdy homotopy concept as a department of arithmetic in its personal correct with purposes in different fields of arithmetic. it's a first step towards making solid homotopy idea a device worthwhile in lots of disciplines of arithmetic.
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Extra resources for Axiomatic stable homotopy theory
Example text
Namely, if we divide G by the maximal compact subgroup flcfi then, by the compactness of H, the spact V = G/H admits a G-invariant metric g. Such a metric is complete (this is elementary) and by a well known theorem of E. Cartan K(g) < 0. This curvature is strictly negative if and only if r a n k ^ G — 1 , and K(g) is constant if and only if G is locally isomorphic to O(n, 1). The compact manifolds V covered by V are associated to discrete subgroups re G which are usually produced by arithmetic constructions.
The sectional curvature K(v) of V c: 1ft3 equals the Jaeobian of the Gauss map V —> Ss at v, or equivalently to the •product of the principal curvatures {eienvahies of TIy or of the shape operator A) at v. Of course, the proof is trivial by the standards of the modern infinitesimal caulculus. Yet, the major consequence of the theorem looks as remarkable as it appeared 200 years ago: the Jaeobian of the Gauss map does not change if we bend V in IR^, that is if we apply a deformation preserving the length of the curves in V.
Furthermore, if dc(W) develops a self-inter section without focal points, then V~ becomes locally represented as an intersection of smooth convex subsets and so again it is convex. Then it is easy to believe in the convexity at the focal points aa well as these are just « infinitesimal » double points (vanishing of the differential of a map at a tangent vector i 6 T(W) brings together the « infinitely closed points » corresponding to the « two ends » of T). SIGN AND GEOMETRIC MEANING OF CURVATURE 53 To make the above rigorous, one may use a piecewise smooth approximation (compare § 0) of convex hypersurfaces (and subsets) as in Fig.