{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

mardesic - Mathematical Communications 2(1997 112 1 Thirty...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Mathematical Communications 2 (1997), 1–12 1 Thirty years of shape theory Sibe Mardeˇ si´c 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 Saˇ zetak . Trideset godina teorije oblika. U ˇ clanku je prikazan razvitak teorije oblika od njenog osnivanja od strane K. Borsuka prije 30 godina do danaˇ snjih dana. Kao motivacija za uvod¯enje teorije ob- lika opisani su neki nedostaci teorije homotopije za prostore nepravilnog lokalnog ponaˇ sanja. Posebna paˇ znja dana je doprinosima teoriji oblika zagrebaˇ cke topoloˇ ske grupe. Kljuˇ cne rijeˇ 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, Bijeniˇ cka cesta 30, HR-10 000 Zagreb, Croatia, e-mail: [email protected]
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2 S. Mardeˇ si´c Borsuk’s 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 (ANR’s), 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 π n ( f ) : π n ( X ) π n ( Y ) of all homotopy groups, is a homotopy equivalence. This theorem does not hold for metric continua.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern