Abstract
The 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.
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© 2009 Birkhäuser Verlag AG
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(2009). Algebraic Topology Background. In: Stable Homotopy Around the Arf-Kervaire Invariant. Progress in Mathematics, vol 273. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9904-7_1
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DOI: https://doi.org/10.1007/978-3-7643-9904-7_1
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-9903-0
Online ISBN: 978-3-7643-9904-7
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