Generalized cohomology operations and 𝐻-spaces of low rank

of the Steenrod squares for the Q2-cohomology rings of finite complexes whose integral homology is free of 2-torsion, where Q2 is the ring of rationals with odd denominators. These are not cohomology operations as they only satisfy generalized naturality properties, but under favourable conditions they provide information which it does not seem possible to obtain using either primary or secondary cohomology operations. The treatment of these generalized cohomology operations given in this note is not complete; only those properties are considered which are needed for the immediate applications. In later papers the author hopes to extend both the theory and the applications of these generalized operations as indicated in [13]. All the additive and several of the multiplicative results given in this paper have quite precise analogues for odd primes. These are not considered here, as the most elegant treatment of the corresponding generalized reduced power operations for odd primes involves the use of theorems of J. F. Adams which have not yet been published. I would like to thank Professor J. F. Adams and Dr. I. M. James for the considerable assistance they have given in the preparation of this paper, which essentially formed a part of the author's doctoral thesis.