View source for Shape theory

The fundamental notions of [[Homotopy|homotopy]] theory can be successfully applied only to spaces whose local behaviour is sufficiently regular, e.g. to manifolds, polyhedra, CW-complexes, and ANR spaces. Shape theory is a modification of homotopy theory designed to give satisfactory results for arbitrary topological spaces, especially for metric compacta. When restricted to spaces with regular local behaviour, shape theory coincides with homotopy theory; therefore it can be viewed as the appropriate extension of homotopy theory to general spaces.

Many constructions in topology lead naturally to spaces with bad local behaviour even when one initially considers manifolds. Standard examples include fibres of mappings, sets of fixed points, remainders of compactifications, attractors of dynamical systems, spectra of operators, or boundaries of certain groups. In all these areas, shape theory has proved useful. In particular, it has applications in the study of cell-like mappings, approximate fibrations and shape fibrations, imbeddings of compacta in Euclidean spaces and in exact homology theories.

Shape theory became a separate area of topology when K. Borsuk defined the shape category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s1201301.png" /> of compact metric spaces and the shape functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s1201302.png" /> defined on the homotopy category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s1201303.png" /> (of metric compacta) [[#References|[a2]]]. The shape category was soon extended to arbitrary topological spaces [[#References|[a17]]]. The standard approach to the construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s1201304.png" /> consists in replacing spaces by admissible inverse systems of polyhedra or ANR spaces, called expansions, and in developing a homotopy theory of such systems. In the case of compact Hausdorff spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s1201305.png" /> (cf. also [[Compact space|Compact space]]), as an expansion one can use any inverse system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s1201306.png" /> whose limit is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s1201307.png" />, [[#References|[a14]]]. In the shape category, objects are topological spaces (cf. [[Topological space|Topological space]]), while the morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s1201308.png" /> are given by morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s1201309.png" /> between polyhedral expansions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013011.png" />, respectively, which belong to the category pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013012.png" />, associated with the homotopy category [[#References|[a8]]].

One of the first successful applications of shape theory is Fox's theory of overlays, a modification of covering space theory [[#References|[a7]]]. While the classical classification theorem of covering spaces requires local connectedness and semi-local <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013013.png" />-connectedness of the base space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013014.png" />, the corresponding result in shape theory is valid for all metric spaces with covering spaces replaced by overlays [[#References|[a7]]]. The role of the [[Fundamental group|fundamental group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013015.png" /> is taken up by the fundamental pro-group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013016.png" />. This is the inverse system of groups obtained from a polyhedral expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013017.png" /> upon application of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013018.png" />. Other classical theorems of homotopy theory also have their shape-theoretic versions. In particular, this includes the theorems of J.H.C. Whitehead, W. Hurewicz and S. Smale. The statements of these results also use pro-groups, in particular the homology and the homotopy pro-groups. In the case of the Whitehead theorem, a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013019.png" /> between pointed finite-dimensional topological spaces is a shape equivalence, i.e., an isomorphism of pointed shape, if and only if it induces isomorphisms of all homotopy pro-groups. In contrast to the classical theorem, which refers to CW-complexes (cf. also [[CW-complex|CW-complex]]), there are no such restrictions in the shape-theoretic version [[#References|[a16]]], [[#References|[a12]]], [[#References|[a4]]].

In 1972, T.A. Chapman considered compacta <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013020.png" /> which are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013022.png" />-imbedded in the Hilbert cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013023.png" />, i.e., have the property that there exist mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013024.png" /> that are arbitrarily close to the identity while their image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013025.png" /> misses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013026.png" />. Chapman's complement theorem asserts that two compacta <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013028.png" />, imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013029.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013030.png" />-sets, have the same shape if and only if their complements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013032.png" /> are homeomorphic. There also exist finite-dimensional complement theorems, where the ambient space is the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013033.png" />. Compacta <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013035.png" /> are required to be  "nicely"  imbedded, i.e., satisfy the inessential loops condition and satisfy appropriate dimensional and shape <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013037.png" />-connectedness conditions [[#References|[a10]]].

Chapman's work also led to the discovery of the strong shape category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013038.png" /> and the strong shape functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013039.png" />. D.A. Edwards and H.M. Hastings established an isomorphism between the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013040.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013042.png" />-sets of the Hilbert cube and the proper homotopy category of their complements [[#References|[a6]]]. Strong shape theory has a richer structure than the usual shape theory, i.e., there exists a forgetful functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013043.png" /> (cf. also [[Functor|Functor]]) which relates the two theories. The strong shape category occupies an intermediate place between the homotopy category and the usual shape theory, i.e., the shape functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013044.png" /> admits a factorization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013045.png" />.

The construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013047.png" /> for arbitrary topological spaces requires a careful choice of admissible expansions, the so-called strong expansions [[#References|[a13]]]. Moreover, the correct homotopy theory of inverse systems is coherent homotopy theory [[#References|[a1]]], [[#References|[a11]]].

The following result is a concrete example of successful applications of shape theory. A finite-dimensional compactum imbeds in a (differentiable) manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013048.png" /> as an attractor of a (smooth) [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120130/s12013049.png" /> if and only if it has the shape of a compact polyhedron [[#References|[a9]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.M. Boardman,   R.M. Vogt,   "Homotopy invariant algebraic structures on topological spaces" , ''Lecture Notes Math.'' , '''347''' , Springer  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Borsuk,   "Concerning homotopy properties of compacta"  ''Fund. Math.'' , '''62'''  (1968)  pp. 223–254</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T.A. Chapman,   "On some applications of infinite-dimensional manifolds to the theory of shape"  ''Fund. Math.'' , '''76'''  (1972)  pp. 181–193</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Dydak,   "The Whitehead and the Smale theorems in shape theory"  ''Dissert. Math.'' , '''156'''  (1979)  pp. 1–55</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Dydak,   J. Segal,   "Shape theory: An introduction" , ''Lecture Notes Math.'' , '''688''' , Springer  (1978)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D.A. Edwards,   H.M. Hastings,   "Čech and Steenrod homotopy theories with applications to geometric topology" , ''Lecture Notes Math.'' , '''542''' , Springer  (1976)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  R.H. Fox,   "On shape"  ''Fund. Math.'' , '''74'''  (1972)  pp. 47–71</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Grothendieck,   "Technique de descentes et théorèmes d'existence en géométrie algébrique II"  ''Sém. Bourbaki'' , '''12'''  (1959/60)  pp. Exp. 190–195</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  B. Günther,   J. Segal,   "Every attractor of a flow on a manifold has the shape of a finite polyhedron"  ''Proc. Amer. Math. Soc.'' , '''119'''  (1993)  pp. 321–329</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  I. Ivanšić,   R.B. Sher,   G.A. Venema,   "Complement theorems beyond the trivial range"  ''Illinois J. Math.'' , '''25'''  (1981)  pp. 209–220</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  Ju.T. Lisica,   S. Mardešić,   "Steenrod–Sitnikov homology for arbitrary spaces"  ''Bull. Amer. Math. Soc.'' , '''9'''  (1983)  pp. 207–210</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  S. Mardešić,   "On the Whitehead theorem in shape theory I"  ''Fund. Math.'' , '''91'''  (1976)  pp. 51–64</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  S. Mardešić,   "Strong expansions and strong shape theory"  ''Topology Appl.'' , '''38'''  (1991)  pp. 275–291</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  S. Mardešić,   J. Segal,   "Shapes of compacta and ANR-systems"  ''Fund. Math.'' , '''72'''  (1971)  pp. 41–59</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  S. Mardešić,   J. Segal,   "Shape theory" , North-Holland  (1982)</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  K. Morita,   "The Hurewicz and the Whitehead theorems in shape theory"  ''Reports Tokyo Kyoiku Daigaku Sec. A'' , '''12'''  (1974)  pp. 246–258</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  K. Morita,   "On shapes of topological spaces"  ''Fund. Math.'' , '''86'''  (1975)  pp. 251–259</TD></TR></table>