Shape theory (mathematics)

Shape theory is a branch of topology, which provides a more global view of the topological spaces than homotopy theory. The two coincide on compacta dominated homotopically by finite polyhedra. Shape theory associates with the Čech homology theory while homotopy theory associates with the singular homology theory.

Background

Shape theory was reinvented, further developed and promoted by the Polish mathematician Karol Borsuk in 1968. Actually, the name shape theory was coined by Borsuk.

Warsaw Circle

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.

The Warsaw Circle.

It has homotopy groups isomorphic to those of a point, but is not homotopy equivalent to a point—instead, the Warsaw circle is shape-equivalent to a circle (one dimensional sphere). Whitehead's theorem does not apply to Warsaw circle because it is not a CW complex.

Remark: to be precise if a bit pedantic, a point above stands for a one point space.

Development

Borsuk's shape theory was generalized onto arbitrary (non-metric) compact spaces, and even onto general categories, by Włodzimierz Holsztyński in year 1968/1969, and published in Fund. Math. 70 , 157-168, y.1971 (see Jean-Marc Cordier, Tim Porter, (1989) below). This was done in a continuous style, characteristic for the Čech homology rendered by Samuel Eilenberg and Norman Steenrod in their monograph Foundations of Algebraic Topology . Due to the circumstance, Holsztyński's paper was hardly noticed, and instead a great popularity in the field was gained by a much less advanced (more naive) paper by Sibe Mardešić and Jack Segal, which was published a little later, Fund. Math. 72, 61-68, y.1971. Further developments are reflected by the references below, and by their contents.

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. 
• shape theory in nLab
• 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.
• D.A. Edwards and H. M. Hastings, Čech Theory: its Past, Present, and Future, Rocky Mountain Journal of Mathematics, Volume 10, Number 3, Summer 1980
• D.A. Edwards and H. M. Hastings, (1976), [www.math.uga.edu/%7Edavide/Cech_and_Steenrod_Homotopy_Theories_with_Applications_to_Geometric_Topology.pdf Čech and Steenrod homotopy theories with applications to geometric topology], Lecture Notes in Maths. 542, Springer-Verlag.
• 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.