Search

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)

Duality is a fundamental symmetry principle in mathematics, appearing in linear algebra, functional analysis, geometry and topology. We describe here one very important aspect of duality which links geometry and topology. Descartes defined the angular deficiency Δ of a convex rectilinear surface hemeomorphic to the sphere and showed that Δ = 4 π . Grunbaum and Shepard recently defined a dual geometric invariant Δ′ for such a surface and showed that Δ ′ = 4 π . We give a combinatorial interpretation of Δ for any (two-dimensional) polyhedron and deduce, from general duality considerations, an interpretation of Δ′. It then follows that Δ = Δ ′ for any closed rectilinear surface and that Δ ′ = 2 πχ , where χ is the famous Euler characteristic ( χ = V − E + F ), for any polyhedron. This yields a new insight into the relation Δ = 2 πχ , proved by the authors for any closed rectilinear surface (see Ref. [2]).

The Eckmann-Hilton (injective) homotopy category of modules over a ring A is equivalent to a certain full subcategory of the cochain homotopy category of cochain complexes of modules over A. 0. Introduction. Eckmann [1] and Hilton [2, 3] reported separately the results of their joint work on a homotopy theory of modules, constructed by analogy with the homotopy theory of pointed topological spaces. In retrospect it is perhaps surprising that the simple result reported here appears not to have been noticed, although the constituent facts have long been well known, and the proof is obvious. The result may be interpreted this way. In topology, homology groups are homotopy invariant-that is, the homomorphism f*: HX -+ HY induced by a continuous function f: X -+ Y depends only on the homotopy class of f. If the analogy referred to above has validity, surely a similar statement should be true in module theory. Corollary 2 below asserts that this is so, provided a small modification is made in dimension 0. 1. An equivalence of categories. Let A be an abelian category with sufficiently many injectives, and let HA denote the injective homotopy category of A [2]. Recall that two morphisms 4, 4: A -+ A' in A are (injectively) homotopic, or i-homotopic, if 4 4 extends to some injective container of A. Of course, 4 4 then extends to any container of A. We will simply write 4 , for the relation of i-homotopy. Now let IRA be the category whose objects are injective resolutions of objects of A and whose arrows are morphisms of exact sequences: d do d, A >4 IO 1 ...

I have chosen this topic for its appropriateness both to the theme of the conference—new directions in applied and computational mathematics—and to the occasion on which we honor Gail Young. For the new directions, with which other speakers will be dealing more directly in their contributions, carry strong implications for the teaching of mathematics at graduate and undergraduate levels—indeed, at all levels—so that it is fitting that we pay attention now to those implications. The organizers have, of course, taken note of this aspect of the conference theme, in placing on our agenda, as its final item, a panel discussion on the general issue of the shape of a new mathematics curriculum. I will confine myself to one aspect of the implications for us as teachers of these new directions; and I promise that my remarks today will be neither repeated nor contradicted when I reappear before you as a panelist tomorrow.

1. Across the country, and beyond the borders of the United States, the cry is being heard that we mathematicians should be concerning ourselves more, both in our research and in our teaching, with applications of mathematics. It is being argued that we have been overemphasizing mathematics itself, the autonomous discipline of mathematics, at the expense of due attention to its usefulness, to its role in science, in engineering, in the conduct of modern society. Some put it crudely—there is too much “ pure mathematics,” too little “ applied mathematics.”

Those mathematicians who have reached, shall we say, a mature stage in their careers find themselves today having to do a great deal of writing. If they remain active in research, then there are research articles, survey articles, texts of invited addresses to be drafted. Moreover, if they seek financial support for their research, then there are proposals to be formulated to potential funding agencies. If they have established good relationships with colleagues and students, then there are a multitude of letters of recommendation to be written in the Utopian hope that these letters might help young colleagues to obtain jobs in the contracting academic market. Over and above this, it may be that a book is being written or that lecture notes have to be prepared for a special course.