By F. Oort

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M. James, Reduced product spaces, Ann. of Math. 62 (1955), 170-197. M. C. Whitehead, The homotopy theory of sphere bundles over spheres: I, Proc. London Math. Soc. 4 (1954), 196-218. [12] I. Kozma, Some relations between semigroups of polyhedra, Proc. Amer. Math. Soc. 39 (1973), 388-394. [13] W. Magnus, Ueber Beziehungen zwischen hoheren Kommutatoren, Journal fur Math. (Crelle) 177 (1937), 105-115. [14] G. Mislin, Cancellation properties of H-spaces, Comment. Math. Helv. 49 (2) 1974, 195-200. [15] G.

The addition in G(N) is then defined by constructing the pushout square N 4- The group L is a representative of the sum in G(N) of the genus class of H and the genus class of K . The abelian group defined in this way can for example be used to estimate the size of G(N) . Stammbach: The work of Peter Hilton in algebra 44 Of course Peter Hilton's mathematical work continues. We certainly shall see in the future, as we have in the past, a lot of significant papers carrying that special trade mark: Peter Hilton.

An axiomatic setting could be developed but would probably obscure the simplicity of the ideas. We shall use very l i t t l e beyond fibre and cofibre sequences. Let XAY be the smash product X x Y/XvY and let F(X,Y) be the function space of based maps X -• Y. The source of duality is the adjunction homeomorphism (1) F(XAY,Z) 2 F(X,F(Y,Z)). Let CX - XAl, LX = XAS 1 , PX = F(I,X), and fiX - FCS^X), where I has basepoint 1 in forming CX and 0 in forming PX. For a based map f :X -• Y, l e t Cf - Y