CHAPTER 1 Introduction In this part we study spaces of type Q,B7r£ in terms of some of their algebraic invariants. In particular we show that if n is a finite p-perfect group, then the integral homology and stable homotopy of QBTT^ have exponents. The exponents that we find in both cases are given by the order of the Sylow p-subgroup of 7r. 1.1. Statement of Results THEOREM 1.1.1. Let n be a finite p-perfect group of order pr m, (ra,p) = 1. Then pr-H*(nB7rZ Z{p))=0. It is known that if n is any finite group of order pr ra, (p, ra) = 1, then pr annihilates the reduced p-primary homology of Bn. However let X be an arbitrary 1-connected space whose p-primary homology has an exponent p r . In general one would not expect p r to annihilate the reduced p-primary homology of flX. In fact the only well known examples are spaces of the form ft£Y, where TJY is the reduced suspension of a connected space Y, with a p-primary homology exponent, and spaces of the form BS2n~l{pr}, where p and n are chosen such that S2n~l{pr} is an associative H-space. It is also not hard to observe that if X is a rationally contractible finite CW complex, then the reduced integral homology of QX has a finite exponent. In particular if p is odd and n divides p 1, then it is shown in [35], and more systematically in [17], that 5 2 n _ 1 {p r } ~ QBTT^ where n is a certain finite p-perfect group whose Sylow p-subgroup is cyclic of order p r . We also point out that spaces of type Bn^ where n is a finite group whose order is divisible by p, are neither suspension spaces nor finite complexes, because of the existence of non-trivial products of arbitrary height in H*(B7Tp ¥p) [53]. Theorem 1.1.1 thus provides a large class of spaces with a rather special property. Furthermore, in chapter 5 below, we give examples where the Sylow 2-subgroup of n has order Received by the editor November 10, 1993 3
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