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The notion ofgenus, applied to finitely generated nilpotent groups or to nilpotent spaces of finite type, was introduced by Mislin; he and the author showed how to introduce the structure of a finite abelian group into the genus if the groupN has finite commutator subgroup. An example is given of a complete genusN0,N1,...,Ns−1, which constitute a cyclic group generated byN1, with the additional property that eachNi embeds in its successor as a normal subgroup with quotient cyclic of orderl; of course,Ns−1 embeds inN0. The construction leads to the description of a family of nilpotent spacesX0,X1,...,Xs−1, all in the same genus, no two of the same homotopy type, such that eachXi covers its successor as a cyclicl-sheeted regular covering; of course,Xs−1 coversX0. Herep is a prime,n ≧ 1, ands=pn−1(p−1)/2, whilel is semiprimitive modulepn.

There has long been a debate as to how geometry should be taught in the High School, and many papers contributed at this conference will doubtless be concerned with this important question. My own attitude is based on certain principles which should, I believe, inform one’s approach to the choice of content in a High School mathematics curriculum and to the choice of methods of presentation. A topic is only deserving of inclusion if it enriches the student’s experience by illuminating past or current interests and concerns, or if it is capable of future — but not too distant future — application. The experience referred to may be mathematical or non-mathematical; likewise, the application may be outside mathematics, or it may be the application of a general mathematical procedure to a more specialized mathematical situation. On this criterion, geometry deserves its place as an application of algebra and as a model of the world of experience. It is not, however, obvious that, in an inevitably crowded curriculum, one can justify the inclusion of geometrical material (for example, problems of coordinatization or Hilbertian axiomatics) which, while elegant and deep parts of mathematics, are somewhat sui generis and do not make a strong impact on the rest of mathematics or on the world of scientific knowledge served by mathematics.

The homotopy classification of 1-connected, finite-dimensional H-spaces of rank two, whose study was begun by Curtis-Mislin [1], Hilton-Roitberg [4] and Zabrodsky [10], has been recently completed by Mimura-Nishida-Toda [5]. The emphasis in the papers [1, 4, 10] is on rank two H-spaces without homological 2-torsion. In this case, it is shown in [1, 4, 10] that, aside from the classical examples, the only occurrences of H-spaces are certain of the (total spaces of) principal Sa-bundles over S 7. In fact, if Eko ~ denotes the bundle whose classifying map is k ~o, where tn is the Blakers-Massey generator s, then, from [3, 8], all Eke o except perhaps for k = 2 , 6 do actually admit H-structures, while from [10], E2~ and E6w do not admit H-structures. Among these seven principal bundles, precisely two, namely E,o=Sp(2) and Es,~, admit loop-space structures. For these spaces, we have the non-cancellation phenomenon referred to in the title, namely E~• 3, E,o~Eso,, (1.1)