{{Short description|Branch of topology}} {{Missing information|the theory itself (concepts, methods, results)|date=April 2026}} '''Shape theory''' is a branch of [[topology]] that provides a more global view of the topological spaces than [[homotopy theory]]. The two coincide on [[compact set|compacta]] dominated homotopically by finite [[simplicial complex|polyhedra]]. Shape theory associates with the [[Čech cohomology|Čech homology]] theory while homotopy theory associates with the [[singular homology]] theory.
==Background==
Shape theory was invented and published by [[D. E. Christie]] in 1944; it 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=== [[File:Warsaw Circle.png|thumb|The Warsaw circle]] Borsuk lived and worked in [[Warsaw]], hence the name of one of the fundamental examples of the area, the '''Warsaw circle'''.<ref>"[http://math.ucr.edu/~res/math205B-2012/polishcircle.pdf The Polish Circle and some of its unusual properties]". Math 205B-2012 Lecture Notes, University of California Riverside. Retrieved November 16, 2023. See also the accompanying diagram "[http://math.ucr.edu/~res/math205B-2012/polishcircleA.pdf Constructions on the Polish Circle]"</ref> It is a compact subset of the plane produced by "closing up" a [[topologist's sine curve]] (also called a ''Warsaw sine curve'') with an arc. The [[homotopy group]]s of the Warsaw circle are all [[Trivial group|trivial]], just like those of a point, and so any map between the Warsaw circle and a point induces a [[weak homotopy equivalence]]. However these two spaces are not [[homotopy equivalence|homotopy equivalent]]. So by the [[Whitehead theorem]], the Warsaw circle does not have the homotopy type of a [[CW complex]].
==Historical development==
Borsuk's shape theory was generalized onto arbitrary (non-[[metric space|metric]]) compact spaces, and even onto general categories, by Włodzimierz Holsztyński in year 1968/1969, and published in [[Fundamenta Mathematicae|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{{Clarify|date=March 2018}}, Holsztyński's paper was hardly noticed, and instead a great popularity in the field was gained by a later paper by [[Sibe Mardešić]] and Jack Segal, Fund. Math. '''72''', 61–68, y.1971. Further developments are reflected by the references below, and by their contents.
For some purposes, like [[dynamical system]]s, more sophisticated invariants were developed under the name '''strong shape'''. Generalizations to [[noncommutative geometry]], e.g. the shape theory for [[operator algebra]]s have been found.
== See also ==
* [[List of topologies]]
== References == {{Reflist}} * {{cite journal | first = Sibe | last = Mardešić | author1-link=Sibe Mardešić| title = Thirty years of shape theory | url = http://hrcak.srce.hr/file/2848 | format = [[PDF]] | journal = Mathematical Communications | volume = 2 | year = 1997 | pages = 1–12}} * {{nlab|id=shape%20theory|title=shape theory}} * Jean-Marc Cordier and Tim Porter, (1989), Shape Theory: Categorical Methods of Approximation, Mathematics and its Applications, [[Simon & Schuster|Ellis Horwood]]. Reprinted Dover (2008) * Aristide Deleanu and [[Peter Hilton|Peter John Hilton]], On the categorical shape of a functor, [[Fundamenta Mathematicae]] 97 (1977) 157–176. * Aristide Deleanu and [[Peter Hilton|Peter John Hilton]], Borsuk's shape and Grothendieck categories of pro-objects, [[Mathematical Proceedings of the Cambridge Philosophical Society]] 79 (1976) 473–482. * [[Sibe Mardešić]] and Jack Segal, Shapes of compacta and ANR-systems, [[Fundamenta Mathematicae]] 72 (1971) 41–59 * [[Karol Borsuk]], Concerning homotopy properties of compacta, [[Fundamenta Mathematicae]] 62 (1968) 223–254 * [[Karol Borsuk]], Theory of Shape, Monografie Matematyczne Tom 59, Warszawa 1975. * D. A. Edwards and H. M. Hastings, [http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.rmjm/1250128825&page=record 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), [http://alpha.math.uga.edu/~davide/Cech_and_Steenrod_Homotopy_Theories_with_Applications_to_Geometric_Topology.pdf Čech and Steenrod homotopy theories with applications to geometric topology], [[Lecture Notes in Mathematics]] 542, [[Springer Science+Business Media|Springer-Verlag]]. * Tim Porter, Čech homotopy I, II, [[London Mathematical Society|Journal of the London Mathematical Society]], 1, 6, 1973, pp. 429–436; 2, 6, 1973, pp. 667–675. * J.T. Lisica and [[Sibe 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, [http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1997__38_1_3_0 numdam] * Marius Dădărlat, Shape theory and asymptotic morphisms for C*-algebras, [[Duke Mathematical Journal]], 73(3):687–711, 1994. * Marius Dădărlat and Terry A. Loring, Deformations of topological spaces predicted by E-theory, In Algebraic methods in operator theory, p. 316–327. [[Birkhäuser]] 1994.
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[[Category:Topology]] [[Category:Homotopy theory]]