Solomon Lefschetz

Intersections and transformations of complexes and manifolds

In writing this paper my first objective has been to prove certain formulas on fixed points and coincidences of continuous transformations of manifolds. To this proof for orientable manifolds without boundary is devoted most of the second part, the remainder of which is taken up by a study of product complexes in the sense of E. Steinitz, as they are the foundation on which the proof rests. With suitable restrictions the formulas derived are susceptible of extension to a wider range of manifolds, but this will be reserved for a later occasion. It may be stated that our formulas include and completely generalize the early results due to Brouwer and whatever has been obtained since along the same line.t No such generality would have been possible without that powerful instrument, the product complex. The principle of the method is best explained by means of a very simple example. Letf(x) and so(x) be continuous and uni-valued functions over the interval 0, 1, and let their values on the interval also lie between 0 and 1. It is required to find the number of solutions of f(x) = (x), 0 < x < 1. Graphically the problem is solved by plotting the curvilinear arcs y =(x), y = (x), 0 < x<1 and taking their intersections. A slight modification of the functions may change tlle number of solutions, even make them become infinite in number. However, the difference between the numbers of positive and negative crossings of sufficiently close polygonal approximations to the arcs is a fixed number, their Kronecker index. Its determination is then a partial answer to the question, and indeed seemingly the only possible general answer.

On analytical complexes

1. In his Colloquium Lecturesf one of us outlined a proof of an important theorem regarding the covering of analytic loci by complexes. A proof for algebraic varieties had previously been given by B. van der WaerdenJ and B. 0. Koopman and A. B. Brown§ have recently proved the theorem for analytic loci. The object of this paper is to give a detailed proof along the lines indicated in Topology. 2. We begin with certain general observations! I concerning the nature of a configuration £ (at first complex) represented by an analytic system

Manifolds with a boundary and their transformations

This is the continuation of the paper which appeared in the January, 1926, number of these Transactions.t Its chief object is to extend to an M. with a boundary the results already obtained for transformations of manifolds without boundary. To these already treated in full we devote a few pages chiefly to elucidate and simplify certain points of importance for the extension. We have succeeded in deriving coincidence and fixed points formulas for the two types of transformations that are alone amenable to anything like a general treatment and extended the formulas of this and the preceding paper to transformations between two different manifolds with or without a boundary. As an incidental acquisition there should be pointed out some highly interesting topological propositions obtained in Parts II, III. Of importance also is the fact that by means of ample use of matrices we have been able to put all coincidence formulas of this and the previous paper in very simple and manageable form.

Recent Soviet contributions to ordinary differential equations and nonlinear mechanics

Abstract This report is an appraisal of recent Soviet contributions to differential equations and nonlinear mechanics. It contains a general appraisal of the significance and implications of Soviet research in this field. A somewhat nontechnical description is given of the major areas of research and the significance of individual Soviet contributions. As an appendix to this report we have included a more technical appraisal of the Soviet contributions. A mathematical abstract together with the names of the authors and exact references are given in this appendix of each of the major papers and books available to us in 1958. Papers which appear to be of undoubted practical importance are designated in the list of references by a dagger before the reference number. An asterisk indicates that we have judged the work to be an outstanding scientific contribution. These symbols also follow the reference numbers (enclosed in square brackets) within the text.

On Generalized Manifolds

The object of the present paper is to extend to a larger class of spaces certain results recently obtained for topological manifolds. † The extension consists in replacing the requirement that every point possess a combinatorial cell for neighborhood by certain weaker conditions on the chains through the point. Eoughly speaking they amount to demanding that locally any p-chain be deformable (in a certain very general sense) into one’which does not meet any assigned q-space (= q dimensional space), where p + q < n, the dimension of the manifold. This extension is made in Part III of the present paper. In Part I we take up again, partly as a preparation to the second Part, the homology theory of metric spaces from the standpoint initiated in our Colloquium Lectures Topology, Ch. VII. The notation and terminology are as in our book. ‡

On Separable Spaces

We owe to E. H. Moore the introduction of a very general type of double sequence { Pi} which he called a development. With such sequences there may be associated certain abstract spaces investigated at length by Chittenden and Pitcher with noteworthy results particularly in connection with the problem of metrization. They dealt at considerable length with so-called regular developments of a space 9S. In a regular development the Ps are neighborhoods of 9R and a has a finite range for every i. Our first object in the present paper is to investigate a type of development called normal whose sets are subjected to more stringent conditions of convergency than with the regular type. It turns out, however, that every separable metric space possesses normal developments. As a consequence they seem to be just what is needed for the treatment of many questions on separable metric. Making use in part of normal developments, we have put on a solid basis the theory of the order for separable spaces, and in particular, completely extended to them the fundamental order theorem (Lebesgue order theorem). It was then a simple matter to prove that every separable metric space can be mapped topologically on a compact metric space of the same dimension. As a consequence certain mapping theorems that we have obtained previously for compact spaces2 hold for separable spaces. By means of

Witold Hurewicz, In memoriam

Last September sixth was a black day for mathematics. For on that day there disappeared, as a consequence of an accidental fall from a pyramid in Uxmal, Yucatan, Witold Hurewicz, one of the most capable and lovable mathematicians to be found anywhere. He had just attended the International Symposium on Algebraic Topology which took place during August at the National University of Mexico and had been the starting lecturer and one of the most active participants. He had come to Mexico several weeks before the meeting and had at once fallen in love with the country and its people. As a consequence he established from the very first a warm relationship between himself and the Mexican mathematicians. His death caused among all of us there a profound feeling of loss, as if a close relative had gone, and for days one could speak of nothing else. Witold Hurewicz was born on June 29, 1904, in Lodz, Russian Poland, received his early education there, and his doctorate in Vienna in 1926. He was a Rockefeller Fellow in 1927-1928 in Amsterdam, privaat docent there till 1936 when he came to this country. The Institute for Advanced Study, the University of North Carolina, Radiation Laboratory and Massachusetts Institute of Technology (since 1945) followed in succession. Mathematically Hurewicz will best be remembered for his important contributions to dimension, and above all as the founder of homotopy group theory. Suffice it to say that the investigation of these groups dominates present day topology. Still very young, Hurewicz attacked dimension theory, on which he wrote together with Henry Wallman the book Dimension theory [39], l We come to this book later. The Menger-Urysohn theory, still of recent creation was then in full bloom, and Menger was preparing his book on the subject. One of the principal contributions of Hurewicz was the extension of the proofs of the main theorems to separable metric spaces [2 to 10 ] which required a different technique from the basically euclidean one of Menger and Urysohn. Some other noteworthy results obtained by him on dimension are: (a) A separable metric nspace ( = n dimensional space) may be topologically imbedded in a compact metric n-space [7].

ON ANALYTICAL COMPLEXES

Publisher Summary This chapter discusses an important theorem regarding the covering of analytic loci by complexes. The chapter presents some general observations concerning the nature of a configuration ξ, at first complex, represented by an analytic system in the vicinity of a given point O of ξ, which is taken as the origin throughout for the complex Euclidean space Sn containing ξ. There is a neighborhood of O relative to ξ, consisting of a finite number of algebroid elements, with any one of them having a canonical representation about its center O, in a suitable coordinate system yi.

Differential Equations: Geometric Theory.

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