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On Finite Groups and Homotopy Theory
Ran Levi, University of Heidelberg, Germany
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Memoirs of the American Mathematical Society
1996; 100 pp; softcover
Volume: 118
ISBN-10: 0-8218-0401-4
ISBN-13: 978-0-8218-0401-8
List Price: US$41
Individual Members: US$24.60
Institutional Members: US$32.80
Order Code: MEMO/118/567
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Let \(p\) be a fixed prime number. Let \(G\) denote a finite \(p\)-perfect group. This book looks at the homotopy type of the \(p\)-completed classifying space \(BG_p\), where \(G\) is a finite \(p\)-perfect group. The author constructs an algebraic analog of the Quillen's "plus" construction for differential graded coalgebras. This construction is used to show that given a finite \(p\)-perfect group \(G\), the loop spaces \(BG_p\) admits integral homology exponents. Levi gives examples to show that in some cases our bound is best possible. It is shown that in general \(B\ast _p\) admits infinitely many non-trivial \(k\)-invariants. The author presents examples where homotopy exponents exist. Classical constructions in stable homotopy theory are used to show that the stable homotopy groups of these loop spaces also have exponents.

Readership

Researchers in algebraic topology, and finite group theory and homotopy theory.

Table of Contents

Part 1: The Homology and Homotopy Theory Associated with \(\Omega B\pi ^\wedge _p\)
  • Introduction
  • Preliminaries
  • A model for \(S_\ast \Omega X^\wedge _R\)
  • Homology exponents for \(\Omega B\pi ^\wedge _p\)
  • Examples for homology exponents
  • The homotopy groups of \(B\pi ^\wedge _p\)
  • Stable homotopy exponents for \(\Omega B\pi ^\wedge _p\)
Part 2: Finite Groups and Resolutions by Fibrations
  • Introduction
  • Preliminaries
  • Resolutions by fibrations
  • Sporadic examples
  • Groups of Lie type and \(\mathcal S\)-resolutions
  • Clark-Ewing spaces and groups
  • Discussion
  • References
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