# On Finite Groups and Homotopy Theory

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*Ran Levi*

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.

#### Table of Contents

# Table of Contents

## On Finite Groups and Homotopy Theory

- Contents v6 free
- Abstract vii8 free
- Preface ix10 free
- Acknowledgements xiii14 free
- Part 1: The Homology and Homotopy Theory Associated with ΩBπ[sup(^)[sub(p)] 116 free
- Chapter 1. Introduction 318
- Chapter 2. Preliminaries 722
- Chapter 3. A model for S[sub(*)]ΩX[sup(^)sub(R)] 924
- Chapter 4. Homology Exponents for ΩBπ[sup(^)[sub(p)] 1328
- Chapter 5. Examples for Homology Exponents 2136
- Chapter 6. The Homotopy Groups of Bπ[sup(^)[sub(p)] 2540
- Chapter 7. Stable Homotopy Exponents for ΩBπ[sup(^)[sub(p)] 3146

- Part 2: Finite Groups and Resolutions by Fibrations 4156
- References 97112