Memoirs of the American Mathematical Society 1996; 100 pp; softcover Volume: 118 ISBN10: 0821804014 ISBN13: 9780821804018 List Price: US$44 Individual Members: US$26.40 Institutional Members: US$35.20 Order Code: MEMO/118/567
 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 nontrivial \(k\)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
 ClarkEwing spaces and groups
 Discussion
 References
