On Finite Groups and Homotopy Theory

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.