Topolwy Vol pp Pergamon Press Printed in Great Britain

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Topolwy Vol. 3, pp. 193422. Pergamon Press, 1965. Printed in Great Britain ON THE GROUPS J(X)-111 J. F. ADAMS (Received 25 November 1963) $1. INTRODUClTON THE GENERAL OBJECT of this series of papers is to give means for computing the groups J(X). A general introduction has been given at the beginning of Part I [3]. We recall that in Part II [4] we set up two further groups J'(X) and J”(X); here J'(X) is a “lower bound” for J(X), and we conjecture that J”(X) is an “upper bound” for J(X). The present paper, Part III, has two main objects; the first is to prove the following theorem. THEOREM (1.1). For each finite CW-complex X we have J'(X) = J”(X). The precise sense in which the groups J'(X) and J”(X) are “equal” is the following. Both groups can be defined as quotients of &(X), say J'(X) = &(X)/V(X) J”(X) = &(X)/W(X).