Algebraic Topology Background

Abstract

AbstractThe objective of this chapter is to sketch the historical and technical stable homotopy background which we shall need in the course of this book. § 1 deals with the history of the calculations of stable homotopy groups of spheres (the so-called “stable stems”). §2 describes the framed manifold approach of Pontrjagin and Thom. § 3 introduces the classical stable homotopy category of spectra and § 4 describes the category’s classical Adams spectral sequence. § 5 introduces the Snaith splittings and derives the Kahn-Priddy theorem, which is essential in order to be able to study the stable homotopy groups of spheres via the stable homotopy groups of ℝℙ∞. §6 recapitulates the properties of Steenrod’s cohomology operations and § 7 does the same for the Dyer-Lashof algebra of homology operations. Finally § 8 describes several equivalent formulations of the Arf-Kervaire invariant one problem.KeywordsHopf AlgebraSpectral SequenceSmash ProductStable HomotopyAdams Spectral SequenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.