Shape theory (mathematics)

Shape theory is a branch of the mathematical field of topology. Homotopy theory is not appropriate for space with bad local properties, hence the need for replacement of homotopy theory by a more sophisticated approach. Thus the overall goal of shape theory is to modify the methods and results from homotopy theory for spaces with good local properties, like CW complexes, to more general compact metric spaces or compact Hausdorff spaces with possibly bad local properties.

Shape theory was founded by the Polish mathematician Karol Borsuk in 1968. Borsuk lived and worked in Warsaw, hence the name of one of the fundamental examples of the area, the Warsaw circle. This is a compact subset of the plane produced by "closing up" a topologist's sine curve with an arc. It has homotopy groups isomorphic to those of a point, but is not homotopy equivalent to it; Whitehead's theorem does not apply because the Warsaw circle is not a CW complex.

Borsuk's original shape theory has been replaced by a more systematic approach by inverse systems, pioneered by Sibe Mardešić, and independently, by Timothy Porter. In abstract terms, one starts with a dense subcategory of good objects, and approximates general objects by inverse systems of good objects in the best way in the sense of certain universality property. Thus the object is replaced by pro-object in dense category in appropriate way.

For some purposes, like dynamical systems, more sophisticated invariants were developed under the name strong shape. Generalizations to noncommutative geometry, e.g. the shape theory for operator algebras have been found.

References

• Mardešić, Sibe (1997). "Thirty years of shape theory" (PDF). Mathematical Communications. 2: 1–12.

• nlab: shape theory

• Jean-Marc Cordier, Tim Porter, (1989), Shape Theory: Categorical Methods of Approximation, Mathematics and its Applications, Ellis Horwood. Reprinted Dover (2008)

• A. Deleanu and P.J. Hilton, On the categorical shape of a functor, Fund. Math. 97 (1977) 157 - 176.

• A. Deleanu, P.J. Hilton, Borsuk's shape and Grothendieck categories of pro-objects, Math. Proc. Camb. Phil. Soc. 79 (1976) 473-482.

• Sibe Mardešić, Jack Segal, Shapes of compacta and ANR-systems, Fund. Math. 72 (1971) 41-59,

• K. Borsuk, Concerning homotopy properties of compacta, Fund Math. 62 (1968) 223-254
• K. Borsuk, Theory of Shape, Monografie Matematyczne Tom 59,Warszawa 1975.

DAVID A. EDWARDS AND HAROLD M. HASTINGS, CECH THEORY: ITS PAST, PRESENT, AND FUTURE,ROCKY MOUNTAIN JOURNAL OF MATHEMATICS, Volume 10, Number 3, Summer 1980 http://rmmc.asu.edu/TO%20DOUGLAS/RMJ/vol10/vol10-3/edw.pdf

• D.A. Edwards and H. M. Hastings, (1976), Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Maths. 542, Springer-Verlag.

http://bib.tiera.ru/dvd56/Edwards%20D.%20A.,%20Hastings%20H.%20M.%20-%20Cech%20and%20Steenrod%20Homotopy%20Theories%20with%20Applications%20to%20Geometric%20Topology%281976%29%28308%29.pdf

• Tim Porter, Čech homotopy I, II, Jour. London Math. Soc., 1, 6, 1973, pp. 429-436; 2, 6, 1973, pp. 667-675.

• J.T. Lisica, S. Mardešić, Coherent prohomotopy and strong shape theory, Glasnik Matematički 19(39) (1984) 335--399.

• Michael Batanin, Categorical strong shape theory, Cahiers Topologie Géom. Différentielle Catég. 38 (1997), no. 1, 3--66, numdam

• Marius Dādārlat, Shape theory and asymptotic morphisms for C*-algebras, Duke Math. J., 73(3):687-711, 1994.
• Marius Dādārlat, Terry A. Loring, Deformations of topological spaces predicted by E-theory, In Algebraic methods in operator theory, p. 316-327. Birkhäuser 1994.