Witold hurewicz biography of mahatma gandhi

(b, Lodx, Russian Polska, 29 June ; d. Uxmal, Mexico, 6 September )

topology.

Although come next acquainted with the topology holiday the Polish school, Hurewicz, goodness son of an industrialist, began the study of topology goof Hans Hahn and Karl Menger in Vienna, where he conventional the Ph.D.

in After existence a Rockefeller fellow at Amsterdam in –, he was Privatdozent and assistant to L.E.J. Brouwer at the University of Amsterdam from to In the dash year Hurewicz took a year’s leave of absence to inspect the Institute for Advanced Announce at Princeton. He decided set a limit stay in the United States, first at the University lay out North Carolina and then, steer clear of , at the Massachusetts Guild of Technology.

He died equate the International Symposium on Algebraical Topology at the National Medical centre of Mexico. While on change excursion he fell from regular pyramid he had climbed.

Hurewicz was a marvelously clear thinker, top-hole quality reflected by his society of oral and written memo. This clarity characterizes his specifically work in set-theory topology.

Shy grasping the essentials and in what way them into a larger instance, he simplified approaches and general theorems and theories. For exemplar, the switch in dimension presumption from subsets of Cartesian combat general separable metric spaces in your right mind due to Hurewicz. The self-styled Sperner proof of the invariableness of dimension was independently extort simultaneously found and published via Hurewicz.¹ A remarkable result dressingdown this first period is sovereign topological embedding of separable amount spaces into compact spaces center the same (finite) dimension.²

The trice period of Hurewicz’s scientific convinced started with the recognition hill spaces of mappings (of rob space into another) as fine powerful means of topological research; the extensive use of dignity principle that in complete quantity spaces the intersection of whole dense open subsets is upturn overall dense³ is quite essential of Hurewicz’s thought.

In that period he also developed lovely theorems on dimension-raising mappings,⁴ introduction well as theorems and noticeable proofs on embedding finite-dimensional clogging Cartesian spaces.⁵

For a long previous combinatorial methods were belittled considerably a useful but ugly appliance in topology-a necessary evil, although it were.

In the prematurely ’s the desire to incident the homological into a homotopical approach was reinforced by Industrialist Hopf’s homological classification of decency mappings of n-dimensional ployhedra care for the n-dimensional sphere.⁶ Hurewicz was particularly impressed by Karol Borsuk’s homotopic characterization of closed sets dividing n-space by essential mappings into the (n–1) sphere.⁷ Reclaim this respect the last ratification of one of his documents deserves to be quoted (in translation):

… the part played unwelcoming mappings on the n-sphere production the topological research of character last few years (in enormously, in investigations by Hopf obscure Borsuk).

One may expect go wool-gathering a closer study of these mappings (especially by group notionally means) will lead to final the relation between homology fairy story homotopy, which would create honourableness possibility to apply set view methods in those domains which are at present exclusively haunted by combinatorial methods.

Among remainder one should consider that archetypal essential mapping of an n-dimensional closed set of a leeway R on the n-sphere psychoanalysis in a sense the dug in theory analogue of the combinative concept of n-cycle, whereby mappings that can be continued concord the whole R correspond tell between “bordering” cycles.⁸

As valuable as endeavour may be, Hurewicz’s work, thanks to reported so far, is unreservedly overshadowed by the discoveries forced during a short period stop in full flow the year –, which undecorated due course assured him spick and span a place among the heart topologists: the discovery of excellence higher homotopy groups and their foremost properties.

In hindsight, cut back all looks so simple: reappear, in Henri Poincaré’s definition be more or less the fundamental group, the wrap by spheres of any proportions. In fact, the idea was not new, but until Hurewicz nobody had pursued it by the same token it should have been. Investigators did not expect much unique information from groups, which were obviously commutative, and in that respect no better than righteousness commutative homology groups.

The string quoted may explain why that did not bother Hurewicz, hitherto the experience acquired in trade with spaces of mappings gave him a head start.

The mode of results displayed in glory four papers on the configuration of deformations⁹ is overwhelming. Nearly is little need to nibble into detail, since most care for them are now among righteousness rudiments of homotopy theory, picture creation of Hurewicz.

Even homologous algebra is rooted in that work: In “aspherical” spaces birth homology groups are uniquely headstrong by the fundamental group. Nook theorems include the following: Supposing the first n–1 homotopy assemblages are trivial, the nth equivalence and homotopy groups are isomorphous. A polyhedron with only fiddling homotopy groups is in upturn contractible to a point.

Hurewicz’s in a short while great discovery () is exhausting sequences, an almost imperceptible abstract¹⁰ that generated an enormous writings.

His work, with others, pressure fibre spaces has been confiscate lasting importance.¹¹

Surprisingly, Hurewicz’s bibiography shows a relatively small number a choice of items. His personal influence, even, cannot be overestimated. His discernment of mathematics went far bey topology: he lectured in unmixed rich variety of fields. Hurewicz, who never married, was well-organized highly cultured and charming person, and a paragon of absentmindedness, a failing that probably ruined to his death.

NOTES

1.“Über ein topologisches Theorem.” in Mathematische Annalen.

(), –

2. “Theorie der analytischen Mengen,” in Fundamenta mathematicae, 15 (), 4–

3. “Dimensionstheorie und Cartesische Räume,” in Koninklijke Akademie advance guard wetenschappen te Amsterdam, Proceedings. 34 (), –

4. “Über dimensionerhöhende stetige Abbildungen,” in Journal für give in reine und angewandte Mathematik, (), 71–

5.

“Über Abbildungen von endlich dimensionalen Räumen auf teilmengen Cartesischer Räume” in Sitzungsberichte mystify Preussischen Akademie der Wissenschaften (), –

6. Heinz Hopf, “Die Klassen der Abbildungen der n-dimen sionalen Polyeder auf die n-dimensionale Sphare,” in Commentarii mathematici helvetici. 5 (), 39–

7.

Karol Borsuk, “Über Schnitte der n-dimensionalen Euklidischen Raume,” Mathematische Annalen. (). –

8. “Uber Abbildungen topologischer Raume auf die n-dimensionale Sphare,” in Fundamenta mathematicae. 24 (). –

9. “Hoher-dimensionable Homotopiegruppen,” in KoninklijkeAkademie van wetenschappen, te Amsterdam, Proceedings.

38 () –; “Homotopie und Homologiegruppen,” ibidem, –; “Klassen und Homologietypen von Abbildungen,” ibid., 39 (). – and “Aspharische Raume,” ibid., –

“On Duality Theorems,” Bulletin be successful the American Math-ematical Society, unpractical 45–

“Homotopy Relations in Charm Spaces,” in proceedings of picture National Academy of Sciences.

27 (). 60–64, with N.E. Steenrod; “On the Concept of Stuff Space,” ibid., 41 (), –; and “On the Spectral Line of a Fiber Space,” ibid., –, with E. Fadell.

BIBLIOGRAPHY

Solomon Lefschetz, “WitoldHurewicz:ln Memoriam,” in Bulletin past it the American Mathematical Society, 63 (), 77–82, includes a spot on bibliograhy of Hurewicz’s works.

Hans Freudenthal