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On Finite Groups and Homotopy Theory

Ran Levi University of Heidelberg
Available Formats:
Electronic ISBN: 978-1-4704-0146-7
Product Code: MEMO/118/567.E
List Price: $44.00 MAA Member Price:$39.60
AMS Member Price: $26.40 Click above image for expanded view On Finite Groups and Homotopy Theory Ran Levi University of Heidelberg Available Formats:  Electronic ISBN: 978-1-4704-0146-7 Product Code: MEMO/118/567.E  List Price:$44.00 MAA Member Price: $39.60 AMS Member Price:$26.40
• Book Details

Memoirs of the American Mathematical Society
Volume: 1181996; 100 pp
MSC: Primary 55;

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.

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

• Chapters
• Part 1. The homology and homotopy theory associated with $\Omega B\pi _p^\wedge$
• 1. Introduction
• 2. Preliminaries
• 3. A model for $S_*{\Omega }X^\wedge _R$
• 4. Homology exponents for ${\Omega }B\pi ^\wedge _p$
• 5. Examples for homology exponents
• 6. The homotopy groups of $B\pi ^\wedge _p$
• 7. Stable homotopy exponents for ${\Omega }B\pi ^\wedge _p$
• Part 2. Finite groups and resolutions by fibrations
• 1. Introduction
• 2. Preliminaries
• 3. Resolutions by fibrations
• 5. Groups of Lie type and $\mathcal {S}$-resolutions
• 6. Clark-Ewing spaces and groups
• 7. Discussion
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Volume: 1181996; 100 pp
MSC: Primary 55;

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.

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

• Chapters
• Part 1. The homology and homotopy theory associated with $\Omega B\pi _p^\wedge$
• 1. Introduction
• 2. Preliminaries
• 3. A model for $S_*{\Omega }X^\wedge _R$
• 4. Homology exponents for ${\Omega }B\pi ^\wedge _p$
• 5. Examples for homology exponents
• 6. The homotopy groups of $B\pi ^\wedge _p$
• 7. Stable homotopy exponents for ${\Omega }B\pi ^\wedge _p$
• Part 2. Finite groups and resolutions by fibrations
• 1. Introduction
• 2. Preliminaries
• 3. Resolutions by fibrations
• 5. Groups of Lie type and $\mathcal {S}$-resolutions