mardesic - Mathematical Communications 2 (1997), 112 1...

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Unformatted text preview: Mathematical Communications 2 (1997), 112 1 Thirty years of shape theory Sibe Marde sic Abstract . The paper outlines the development of shape theory since its founding by K. Borsuk 30 years ago to the present days. As a mo- tivation for introducing shape theory, some shortcomings of homotopy theory in dealing with spaces of irregular local behavior are described. Special attention is given to the contributions to shape theory made by the Zagreb topology group. Key words: homotopy, shape, homotopy groups, homotopy progroups Sa zetak . Trideset godina teorije oblika. U clanku je prikazan razvitak teorije oblika od njenog osnivanja od strane K. Borsuka prije 30 godina do dana snjih dana. Kao motivacija za uvodenje teorije ob- lika opisani su neki nedostaci teorije homotopije za prostore nepravilnog lokalnog pona sanja. Posebna pa znja dana je doprinosima teoriji oblika zagreba cke topolo ske grupe. Klju cne rije ci: homotopija, oblik, grupe homotopije, progrupe ho- motopije AMS subject classifications: 55P55, 01A60 Received February 12, 1997 It is generally considered that the theory of shape was founded in 1968 when the Polish topologist Karol Borsuk (1905-1982) published his well-known paper on the homotopy properties of compacta [4]. However, he submitted the paper on Feb- ruary 2, 1967 and spoke about his result at the Symposium on Infinite-dimensional Topology, held in Baton Rougeu, Louisiana, USA from March 27 to April 1, 1967. This means that we are only a few months away from the thirtieth birthday of shape theory. Borsuk also presented his results and ideas at the International Symposium on Topology and its Applications, held in Hercegnovi from August 25 to 31, 1968 (see [5]). This was the first time that he used the suggestive term shape . Several topologists from Zagreb attended that conference and thus, had the opportunity to learn about shape theory shortly after it was inaugurated. Expanded version of a lecture presented at the Mathematical Colloquium in Osijek, orga- nized by the Croatian Mathematical Society - Division Osijek, November 15, 1996. Department of Mathematics, University of Zagreb, Bijeni cka cesta 30, HR-10 000 Zagreb, Croatia, e-mail: 2 S. Marde sic Borsuks starting point was the observation that many theorems in homotopy theory are valid only for spaces with good local behavior, e.g., manifolds, CW- complexes and absolute neighborhood retracts (ANRs), but fail for spaces like metric compacta. A simple example of this phenomenon is the well-known theo- rem of J.H.C. Whitehead, which asserts that a mapping between connected CW- complexes f : X Y, which induces isomorphisms n ( f ) : n ( X ) n ( Y ) of all homotopy groups, is a homotopy equivalence. This theorem does not hold for metric continua....
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mardesic - Mathematical Communications 2 (1997), 112 1...

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