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Stable Homotopy over the Steenrod Algebra
John H. Palmieri, University of Washington, Seattle, WA
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Memoirs of the American Mathematical Society
2001; 172 pp; softcover
Volume: 151
ISBN-10: 0-8218-2668-9
ISBN-13: 978-0-8218-2668-3
List Price: US$62
Individual Members: US$37.20
Institutional Members: US$49.60
Order Code: MEMO/151/716
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We apply the tools of stable homotopy theory to the study of modules over the mod \(p\) Steenrod algebra \(A^{*}\). More precisely, let \(A\) be the dual of \(A^{*}\); then we study the category \(\mathsf{stable}(A)\) of unbounded cochain complexes of injective comodules over \(A\), in which the morphisms are cochain homotopy classes of maps. This category is triangulated. Indeed, it is a stable homotopy category, so we can use Brown representability, Bousfield localization, Brown-Comenetz duality, and other homotopy-theoretic tools to study it. One focus of attention is the analogue of the stable homotopy groups of spheres, which in this setting is the cohomology of \(A\), \(\mathrm{Ext}_A^{**}(\mathbf{F}_p,\mathbf{F}_p)\). We also have nilpotence theorems, periodicity theorems, a convergent chromatic tower, and a number of other results.

Readership

Graduate students and research mathematicians interested in algebraic topology.

Table of Contents

  • Preliminaries
  • Stable homotopy over a Hopf algebra
  • Basic properties of the Steenrod algebra
  • Chromatic structure
  • Computing Ext with elements inverted
  • Quillen stratification and nilpotence
  • Periodicity and other applications of the nilpotence theorems
  • Appendix A. An underlying model category
  • Appendix B. Steenrod operations and nilpotence in \(\mathrm{Ext}_\Gamma^{**}(k,k)\)
  • Bibliography
  • Index
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