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What is the theory of shape?

Published online by Cambridge University Press:  17 April 2009

Karol Borsuk  and
Jerzy Dydak
Karol Borsuk
Affiliation:
ul Filtrowa 63 m 18, 02-056 Warsaw, Poland.
Jerzy Dydak
Affiliation:
ul Sulmierzycka 6 m 62, 02-139 Warsaw, Poland.
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Abstract

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This expository article on Shape Theory contains the main concepts of this theory with a formulation of the most important results of this theory and also with some open problems. The proofs are omitted, however the article gives references to the hooks and papers, in which the reader can find the proofs.

For simplicity, we formulate several theorems only in the form which clearly gives their geometric sense, even if there are known more general results with rather complicated formulations. The reader more interested in details, can find them in the original papers quoted in the list of references.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 22 , Issue 2 , October 1980 , pp. 161 - 198
Copyright
Copyright © Australian Mathematical Society 1980

References

[1]Александров, П.С. [P.S. Aleksandrov], “О некоторых основных моментах в истории топологии текучего столетия” [Some of the major moments in the history of topology in the current century], Proceedings of the Topology Conference, Kisyniev, 1979.Google Scholar
[2]Александров, П.С. [P.S. Aleksandrov], “Opening Address”, Тuрасnолòскuŭ Сuмnозuум nо общеŭ Тоnолоƨuu uее Прuложекuям [Proceedings of the Symposium on General Topology and Applications], Moscow 1979. to appear.Google Scholar
[3]Александров, П.С., Фелорчук, В.В., [P.S. Aleksandrov and V.V. Fedorchuk], “Основные моменты в развитии теоретико-множествекной топологии” [The main aspects in the developement of set-theoretical topology], Uspehi Mat. Nauk 33 (1978), 3–48; English Translation: Russian Math. Surveys 33 (1978), 1–53.Google ScholarPubMed
[4]Armentrout, Steve, Cellular decompositions of 3-manifolds that yield 3-manifolds (Memoirs of the American Mathematical Society, 107. American Mathematical Society, Providence, Rhode Island, 1971).Google Scholar
[5]Artin, M., Mazur, B., Etale homotopy (Lecture Notes in Mathematics, 100. Springer-Verlag, Berlin, Heidelberg, New York, 1969).CrossRefGoogle Scholar
[6]Ball, B.J., “Alternative approaches to proper shape theory”, Studies in topology, 1–27 (Proc. Conf., University of North Carolina, Charlotte, North Carolina, 1974. Academic Press, New York, London, 1975).Google Scholar
[7]Ball, B.J. and Sher, R.B., “A theory of proper shape for locally compact metric spaces”, Fund. Math. 86 (1974–1975), 163–192.CrossRefGoogle Scholar
[8]Богатый, М. [S. Bogatyĭ], “ О теореме Виеториса в категории гомотоний и одной проблеме Борсука” [The theorem of Vietoris in the homotopy category, and a certain problem of Borsuk], Fund. Math. 84 (1974), 209–228.CrossRefGoogle ScholarPubMed
[9]Borsuk, K., “On some metrizations of the hyperspace of compact sets”, Fund. Math. 41 (1955), 168–202.CrossRefGoogle Scholar
[10]Borsuk, Karol, Theory of retracts (Monografie Matematyczne 44. PWN – Polish Scientific Publishers, Warszawa, 1967).Google Scholar
[11]Borsuk, Karol, “Concerning homotopy properties of compacta”, Fund. Math. 62 (1968), 223–254.CrossRefGoogle Scholar
[12]Borsuk, K., Theory of shape (Lecture Notes Series, 28. Matematisk Institut, Aarhus Universitet, Aarhus, 1971. Reprinted, 1973).Google Scholar
[13]Borsuk, Karol, “On positions of sets in spaces”, Fund. Math. 79 (1973), 141–158.CrossRefGoogle Scholar
[14]Borsuk, Karol, Theory of shape (Monografie Matematyczne 59. PWN – Polish Scientific Publishers, Warszawa, 1975).Google Scholar
[15]Ворсун, Н. [K. Borsuk], Теорuа Шеunоɞ [Theory of Shape] (Mir, Moscow, 1976).Google Scholar
[16]Borsuk, Karol, “On a new shape invariant”, Topology Proceedings, Vol. I, 1–9 (Conf., Auburn Univ., Auburn, Ala., 1976. Mathematics Department, Auburn University, Auburn, Alabamba, 1977).Google Scholar
[17]Borsuk, Karol, “A remark on shape of continua”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 1141–1147.Google Scholar
[18]Borsuk, Karol, “On a metrization of the hyperspace of a metric space”, Fund. Math. 94 (1977), 191–207.CrossRefGoogle Scholar
[19]Borsuk, Karol, “On the Lusternik-Schnirelmann category in the theory of shape”, Fund. Math. 99 (1978), 35–42.CrossRefGoogle Scholar
[20]Borsuk, K., “On decompositions of compacta into compacta of trivial shape”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. (to appear).Google Scholar
[21]Borsuk, Karol, “On shape-regularity of several locally connected continua”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. (to appear).Google Scholar
[22]Borsuk, K. and Holsztyński, W., “Concerning the ordering of shapes of compacta”, Fund. Math. 68 (1970), 107–115.CrossRefGoogle Scholar
[23]Borsuk, K. and Olȩdzki, J., “Remark on the shape–domination”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. (to appear).Google Scholar
[24]Boxer, Laurence and Sher, R.B., “Borsuk's fundamental metric and shape domination”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), 849–853.Google Scholar
[25]Case, J.H. and Chamberlin, R.E., “Characterizations on tree-like continua”, Pacific J. Math. 10 (1960), 73–84.CrossRefGoogle Scholar
[26]Chapman, T.A., “On some applications of infinite-dimensional manifolds to the theory of shape”, Fund. Math. 76 (1972), 181–193.CrossRefGoogle Scholar
[27]Chapman, T.A., “Shapes of finite–dimensional compacta”, Fund. Math. 76 (1972), 261–276.CrossRefGoogle Scholar
[28]Chapman, T.A., “Compact Hilbert cube manifolds and the invariance of Whitehead torsion”, Bull. Amer. Math. Soc. 79 (1973), 52–56.CrossRefGoogle Scholar
[29]Chapman, T.A., “Cell-like mappings”, Algebraic and geometric methods in topology, 230–240 (Conf. Topological Methods in Algebraic Topology, State University of New York, Binghamton, 1973. Lecture Notes in Mathematics, 428. Springer-Verlag, Berlin, Heidelberg, New York, 1974).CrossRefGoogle Scholar
[30]Chapman, T.A., “Simple homotopy theory for ANR's”, General Topology Appl. 7 (1977), 165–174.CrossRefGoogle Scholar
[31]Čerin, Zvonko, “Shape fibrations, F-stability and FR-stability”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 27 (1979), 417–423.Google Scholar
[32]Čerin, Zvonko, “e-calmly regular convergence”, submitted.Google Scholar
[33]Čerin, Zvonko and Šostak, A.P., “Some remarks on Borsuk's fundamental metric”, submitted.Google Scholar
[34]Coram, D.S. and Duvall, P.F. Jr, “Approximate fibrations”, Rocky Mountain J. Math. 7 (1977), 275–288.CrossRefGoogle Scholar
[35]van Dantzig, D., “Ueber topologisch homogene Kontinua”, Fund. Math. 15 (1930), 102–125.CrossRefGoogle Scholar
[36]Demers, Luc, “On spaces which have the shape of C.W.-complexes”, Fund. Math. 90 (1975–1976), 1–9.CrossRefGoogle Scholar
[37]Dold, A., Lectures on algebraic topology (Die Grundlehren der mathematischen Wissenschaften, 200. Springer-Verlag, Berlin, Heidelberg, New York, 1972).CrossRefGoogle Scholar
[38]Duvall, P.F. Jr, and Hush, L.S., “A continuum of dimension n which does not embed up to shape in 2n-space”, Proc. Internat. Conf. Geometric Topology, Warsaw 1978 (to appear).Google Scholar
[39]Dydak, J., “Some remarks concerning the Whitehead theorem in shape theory”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), 437–445.Google Scholar
[40]Dydak, Jerzy, “A simple proof that pointed FANR-spaces are regular fundamental retracts of ANR's”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 55–62.Google Scholar
[41]Dydak, Jerzy, “The Whitehead and Smal theorems in shape theory”, Dissertationes Math. (Rozprawy Mat.) 156 (1979), 1–50.Google Scholar
[42]Dydak, J., Nowak, S. and Strok, M., “On the union of two FANR-sets”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), 485–489.Google Scholar
[43]Dydak, Jerzy, Segal, Jack, Shape theory. An introduction (Lecture Notes in Mathematics, 688. Springer-Verlag, Berlin, Heidelberg, New York, 1978).Google Scholar
[44]Edwards, David A. and Geoghegan, Ross, “The stability problem in shape, and a Whitehead theorem in pro-homotopy”, Trans. Amer. Math. Soc. 214 (1975), 261–277.CrossRefGoogle Scholar
[45]Edwards, David A. and Geoghegan, Ross, “Shapes of complexes, ends of manifolds, homotopy limits and the Wall obstruction”, Ann. of Math. (2) 101 (1975), 521–535.CrossRefGoogle Scholar
[46]Edwards, David A. and Geoghegan, Ross, “Infinite-dimensional Whitehead and Vietoris theorems in shape and pro-homotopy”, Trans. Amer. Math. Soc. 219 (1976), 351–360.CrossRefGoogle Scholar
[47]Engelking, Ryszard, Dimension theory (PWN – Polish Scientific Publishers, Warszawa; North Holland, Amsterdam, Oxford, New York; 1978).Google Scholar
[48]Fox, Ralph H., “On shape”, Fund. Math. 74 (1972), 47–71.CrossRefGoogle Scholar
[49]Geoghegan, Ross, “Elementary proofs of stability theorems in pro-homotopy and shape”, General Topology Appl. 8 (1978), 265–281.CrossRefGoogle Scholar
[50]Geoghegan, Ross and Summerhill, R. Richard, “Concerning the shapes of finite-dimensional compacts”, Trans. Amer. Math. Soc. 179 (1973), 281–292.CrossRefGoogle Scholar
[51]Grothendieck, Alexander, “Technique de descente et théorèmes d'existence en géométrie algébrique. II: Le théorème d'existence en théorie formelle des modules”, Seminaire Bourbaki, 12e année, 1959/60, Fasc. 2, Exp. 195 (Secrétariat Mathématique, Paris, 1960).Google Scholar
[52]Handel, D. and Segal, J., “An acyclic continuum with non-movable suspensions”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 171–172.Google Scholar
[53]Hanner, Olof, “Some theorems on absolute neighborhood retracts”, Ark. Mat. 1 (1952), 389–408.CrossRefGoogle Scholar
[54]Hoillingsworth, John G. and Sher, Richard B., “Closed manifolds are of simple shape”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), 287–290.Google Scholar
[55]Holsztyński, W., “An extension and axiomatic characterization of Borsuk's theory of shape”, Fund. Math. 70 (1971), 157–168.CrossRefGoogle Scholar
[56]Ivanšić, Ivan, “Embedding compacta up to shape”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 471–475.Google Scholar
[57]Kadlof, A., “Remarks on Borsuk's problems concerning the operation of the addition of pointed shapes”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), 1001–1006.Google Scholar
[58]Kadlof, Andrzej, “An example resolving Borsuk's problem concerning the index e(X)”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), 905–907.Google Scholar
[59]Kahn, Daniel S., “An example in Čech cohomology”, Proc. Amer. Math. Soc. 16 (1965), 584.Google Scholar
[60]Keesling, James, “On the shape of torus-like continua and compact connected topological groups”, Proc. Amer. Math. Soc. 40 (1973), 297–302.CrossRefGoogle Scholar
[61]Keesling, James, “An algebraic property of the Čech cohomology groups which provents local connectivity and movability”, Trans. Amer. Math. Soc. 190 (1974), 151–162.Google Scholar
[62]Keesling, James, “Shape theory and compact connected abelian topological groups”, Trans. Amer. Math. Soc. 194 (1974), 349–358.CrossRefGoogle Scholar
[63]Keesling, J., “A non-movable trivial-shape decomposition of the Hilbert cube”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), 997–998.Google Scholar
[64]Kodama, Yukihiro, “On the shape of decomposition spaces”, J. Math. Soc. Japan 26 (1974), 636–646.Google Scholar
[65]Kozlowski, George, “Images of ANR's”, Trans. Amer. Math. Soc. (to appear).Google Scholar
[66]Krasinkiewicz, Józef, “Local connectedness and pointed 1-movability”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25Google Scholar
[67]Krasinkiewicz, J., “Continuous images of continua and 1-movability”, Fund. Math. 98 (1978), 141–164.CrossRefGoogle Scholar
[68]Krasinkiewicz, Józef and Minc, Piotr, “Generalized paths and pointed 1-movability”, Fund. Math. 104 (1979), 141–153.CrossRefGoogle Scholar
[69]Kuratowski, K., “Sur une méthode de métrisation complète de certains espaces d'ensembles compacts”, Fund. Math. 43 (1956), 114–138.CrossRefGoogle Scholar
[70]Kuratowski, K., Topology. Volume I (translated from the French by Jaworowski, J.. Academic Press, New York, London; Państwowe Wydawnictwo Naukowe, Warsaw; 1966). See also: К. Куратовский, Тоnолоƨuя. Tom I (translated from the English by M.Ja. Antonovskiĭ. Izdat. “Mir”, Moscow, 1966).Google Scholar
[71]Kuratowski, K., Topology. Volume II (translated from the French by Kirkor, A.. Academic Press, New York, London; Panstwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw; 1968). See also: К. Куратовский, Тоnолоƨuя. Tom II (translated from the English by M.Ja. Antonovskiĭ. Izdat. “Mir”, Moscow, 1969).Google Scholar
[72]Lacher, R.C., “Cell-like mappings and their generalizations”, Bull. Amer. Math. Soc. 83 (1977), 495–552.CrossRefGoogle Scholar
[73]Mardešić, Sibe, “Shapes for topological spaces”, General Topology Appl. 3 (1973), 265–282.CrossRefGoogle Scholar
[74]Mardešić, Sibe, “On the Whitehead theorem in shape theory I”, Fund. Math. 91 (1976), 51–64.CrossRefGoogle Scholar
[75]Mardešić, Sibe, “On the Whitehead theorem in shape theory II”, Fund. Math. 91 (1976), 93–103.CrossRefGoogle Scholar
[76]Mardešić, Sibe, Rushing, T.B., “Shape fibrations I”, General Topology Appl. 9 (1978), 193–215.CrossRefGoogle Scholar
[77]Mardešić, Sibe, Rushing, T.B., “Shape fibrations II”, General Topology Appl. (to appear).Google Scholar
[78]Mardešić, Sibe and Segal, Jack, “Shapes of compacta and ANR-systems”, Fund. Math. 72 (1971), 41–59.CrossRefGoogle Scholar
[79]Mardešić, Sibe and Segal, Jack, “Equivalence of the Borsuk and the ANR-system approach to shapes”, Fund. Math. 72 (1971), 61–68.CrossRefGoogle Scholar
[80]McMillan, D.R. Jr, “One-dimensional shape properties and three-manifolds”, Studies in topology, 367–381 (Proc. Conf. Univ. North Carolina, Charlotte, North Carolina, 1974. Academic Press, New York, 1975).CrossRefGoogle Scholar
[81]Morita, Kiiti, “The Hurewicz and the Whitehead theorems in shape theory”, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 12 (1974), 264–258.Google Scholar
[82]Morita, Kiiti, “On shapes of topological spaces”, Fund. Math. 86 (1975), 251–259.CrossRefGoogle Scholar
[83]Moszyńska, M., “The Whitehead theorem in the theory of shape”, Fund. Math. 80 (1975), 221–263.CrossRefGoogle Scholar
[84]Nowak, Sławomir, “Some properties of fundamental dimension”, Fund. Math. 85 (1974), 211–227.CrossRefGoogle Scholar
[85]Nowak, Sławomir, “On the fundamental dimension of approximatively 1-connected compacta”, Fund. Math. 89 (1975), 61–79.CrossRefGoogle Scholar
[86]Nowak, Sławomir, “Some remarks concerning the fundamental dimension of the cartesian product of two compacta”, Fund. Math. 103 (1979), 31–41.CrossRefGoogle Scholar
[87]Patkowska, Hanna, “A homotopy extension theorem for fundamental sequences”, Fund. Math. 64 (1969), 87–89.CrossRefGoogle Scholar
[88]Sher, Richard B., “Realizing cell-like maps in euclidean space”, General Topology Appl. 2 (1972), 75–89.CrossRefGoogle Scholar
[89]Sher, R.B., “Property SUV ∞ and proper shape theory”, Trans. Amer. Math. Soc. 190 (1974), 345–356.Google Scholar
[90]Sher, R.B., “Proper shape theory and neighborhoods of sets in Q-manifolds”, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 23 (1975), 271–276.Google Scholar
[91]Sher, R.B., “Some alternative notions of position”, Proc. Internat. Conf. Geometric Topology, Warsaw 1978 (to appear).Google Scholar
[92]Siebenmann, L.C., “Approximating cellular maps with homeomorphisms”, Topology 11 (1972), 271–294.CrossRefGoogle Scholar
[93]Sieradzki, Allan J., “Non-uniqueness of homotopy factorizations into irreducible polyhedra”, Fund. Math. 72 (1971), 97–99.CrossRefGoogle Scholar
[94]Spież, S., “On a plane compactum with the maximal shape”, Fund. Math. 78 (1973), 145–156.CrossRefGoogle Scholar
[95]Spież, S., “A majorant for the family of all movable shapes”, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 21 (1973), 615–620.Google Scholar
[96]Taylor, Joseph L., “A counterexample in shape theory”, Bull. Amer. Math. Soc. 81 (1975), 629–632.CrossRefGoogle Scholar
[97]Trybulec, A., “On shapes of movable curves”, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 21 (1973), 727–733.Google Scholar
[98]Trybulec, A., “On the uniqueness of the decomposition of movable shapes into a product of 1-dimensional shapes”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), 69–73.Google Scholar
[99]West, James E., “Mapping Hilbert cube manifolds to ANR's: a solution of a conjecture of Borsuk”, Ann. of Math. (2) 106 (1977), 1–18.CrossRefGoogle Scholar
[100]Whitehead, J.H.C., “Simplicial spaces, nucleii and m-groups”, Proc. London Math. Soc. (2) 45 (1939), 243–327.CrossRefGoogle Scholar
[101]Whitehead, J.H.C., “On the homotopy type of ANR's”, Bull. Amer. Math. Soc. 54 (1948), 1133–1145.CrossRefGoogle Scholar