By Tonny A. Springer and Dirk van Dalen, Tonny A. Springer, Dirk Van Dalen

Hans Freudenthal (1905-1990) used to be a Dutch mathematician, born in Luckenwalde, Germany. His clinical actions have been of a wealthy kind. Enrolling on the collage of Berlin as a scholar within the Twenties, he within the footsteps of his lecturers and have become a topologist, yet with a full of life curiosity in workforce thought. After a protracted trip in the course of the realm of arithmetic, engaged on just about all topics that drew his curiosity, he became towards the sensible and methodological problems with the didactics of arithmetic. the current Selecta are dedicated to Freudenthal's mathematical oeuvre. They comprise a range of his significant contributions, together with his basic contributions to topology akin to the root of the speculation of ends (in the thesis of 1931) in addition to the advent (in 1937) of the suspension and its use in balance effects for homotopy teams of spheres. In workforce concept there's paintings on topological teams (of the Nineteen Thirties) and on a variety of elements of the idea of Lie teams, reminiscent of a paper on automorphisms of 1941. From the later paintings of the Fifties and Nineteen Sixties, papers on geometric features of Lie concept (geometries linked to extraordinary teams, area difficulties) were integrated. Freudenthal's versatility is additional established via decisions from his foundational and old paintings: papers on intuitionistic common sense and topology, a paper on axiomatic geometry reappraising Hilbert's Grundlagen, and a paper summarizing his improvement of Lincos, a common (""cosmic"") language.

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**Extra info for Hans Freudenthal: Selecta (Heritage of European Mathematics)**

**Example text**

He uses this to prove that every free action by C3 on a lens space is equivalent to a free linear action. 1. Consider one parameter families of embedded 2-spheres (for S3) or embedded Heegard tori (for a lens space) which sweep out the total space of the group action. A typical family is a map (S2 x [0,17, S2 x {0,1}) (T2 x [0,11, T2 2. x -r (S3,{x1,x2}) or {0,1}) -> (L(p,q), {c1,c2}) Take such families which are in general position with respect to C3 = {1,A,A2}, that is, Et , AEt , A2Et are in general position for all except a finite number of critical values of t.

F-1 ([y J x = u j=1 X. 3 2j-2 , y B2 n , Zj-1 I n A), Y. = f-1(Cy Y = Bo J=1 J , y 2n n JY. 2j-1 , B = 3=0 vB. J I n A) 2j n , D = J=p U B2 . ;. x; ;e A There are deformation retractions of X*(and Y*) LEMMA D onto D* We construct an equivariant retraction of Xj onto Proof. B . 2j-2 . The argument for Y. is similar. Let a =

Inductively we may suppose that all such have been removed. Consider the one remaining curve C1 in K n AK. be a small tubular neighbourhood of C1 such that N Let N n AN = 0. Write C. R = M - N - (AN) L (A2N)u , so that R - K - AK - A2K = R1 U R2 (disjoint union) when R1 and R2 are A-invariant solid tori. The orbit space is 47 a union of 3 solid tori, which intersect in pairs along boundary annuli. 1 this is enough to establish the existence of a Seifert fibration (with base space S2 and at most 3 exceptional fibres).