Shape theory

The fundamental notions of 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 $\text{Sh}$ of compact metric spaces and the shape functor $S:\text{Ho}\to\text{Sh}$ defined on the homotopy category $\text{Ho}$ (of metric compacta) [a2]. The shape category was soon extended to arbitrary topological spaces [a17]. The standard approach to the construction of $\text{Sh}$ 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 $X$ (cf. also Compact space), as an expansion one can use any inverse system $\mathcal{X}$ whose limit is $X$, [a14]. In the shape category, objects are topological spaces (cf. Topological space), while the morphisms $F:X\to Y$ are given by morphisms $\mathcal{X}\to\mathcal{Y}$ between polyhedral expansions of $X$ and $Y$, respectively, which belong to the category pro-$\text{Ho}$, associated with the homotopy category [a8].

One of the first successful applications of shape theory is Fox's theory of overlays, a modification of covering space theory [a7]. While the classical classification theorem of covering spaces requires local connectedness and semi-local $1$-connectedness of the base space $X$, the corresponding result in shape theory is valid for all metric spaces with covering spaces replaced by overlays [a7]. The role of the fundamental group $\pi_1(X)$ is taken up by the fundamental pro-group of $X$. This is the inverse system of groups obtained from a polyhedral expansion of $X$ upon application of the functor $\pi_1$. 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 $f:(X,*)\to(Y,*)$ 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), there are no such restrictions in the shape-theoretic version [a16], [a12], [a4].

In 1972, T.A. Chapman considered compacta $X$ which are $Z$-imbedded in the Hilbert cube $Q$, i.e., have the property that there exist mappings $f:Q\to Q$ that are arbitrarily close to the identity while their image $f(Q)$ misses $X$. Chapman's complement theorem asserts that two compacta $X$, $Y$, imbedded in $Q$ as $Z$-sets, have the same shape if and only if their complements $Q\backslash X$, $Q\backslash Y$ are homeomorphic. There also exist finite-dimensional complement theorems, where the ambient space is the Euclidean space $\mathbf{R}^n$. Compacta $X$ and $Y$ are required to be "nicely" imbedded, i.e., satisfy the inessential loops condition and satisfy appropriate dimensional and shape $r$-connectedness conditions [a10].

Chapman's work also led to the discovery of the strong shape category $\text{SSh}$ and the strong shape functor $\overline{S}:\text{Ho}\to\text{SSh}$. D.A. Edwards and H.M. Hastings established an isomorphism between the category $\text{SSh}$ of $Z$-sets of the Hilbert cube and the proper homotopy category of their complements [a6]. Strong shape theory has a richer structure than the usual shape theory, i.e., there exists a forgetful functor $E:\text{SSh}\to\text{Sh}$ (cf. also 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 $S$ admits a factorization $S=E\overline{S}$.

The construction of $\text{SSh}$ and $\overline{S}$ for arbitrary topological spaces requires a careful choice of admissible expansions, the so-called strong expansions [a13]. Moreover, the correct homotopy theory of inverse systems is coherent homotopy theory [a1], [a11].

The following result is a concrete example of successful applications of shape theory. A finite-dimensional compactum imbeds in a (differentiable) manifold $M$ as an attractor of a (smooth) flow (continuous-time dynamical system) on $M$ if and only if it has the shape of a compact polyhedron [a9].

References

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