The
Warsaw CircleThe
Warsaw Circle
Although ordinary algebraic topology is a very powerful tool for studying spaces that are locally "nice", even such common or garden spaces as arbitrary compact subsets of the plane may be too locally complicated for it. In 1968, K. Borsuk published the first paper on Shape Theory. This new branch of algebraic topology was designed to handle g eneral compact metric spaces. It used a basically simple idea namely that as any compact metric space is a limit of polyhedra, surely the basic "shape" of a compact metric space should be in some sense the limit of the "shapes" of its approximating system. For instance, the Warsaw circle constructed by completing a segment of a sin 1/x curve with an arc joining the two ends, has the shape of a circle. Shape Theory has now been accepted as a fundamental bridge between algebraic and geometric topology. At Bangor, the research in Shape Theory has centred around the development of Strong Shape Theory. This has developed out of a desire to have a more richly structured shape theory than that developed by Borsuk. Strong shape theory is still in its developmental phase. It has strong connections with proper homotopy theory.
Shape Theory is thus an attempt to apply the highly successful methods of modern algebraic topology to a larger class of spaces than just the manifolds and CW-complexes on which those methods work best. The phenomena studied have often been shown to have close but subtle links with properties of associated algebras of continuous functions, with properties of strange attractors in the theory of dynamical systems, and with fractals. Strong Shape Theory is more highly structured than the original form as it takes into account the natural occurrence of higher homotopy theoretic phenomena (homotopy coherence). It is thus much harder to work with but at the same time shows more promise of providing deep insight into the homotopy theory of the spaces being studied. The hard technical problems arising in Strong Shape Theory can with great difficulty be handled topologically, however the current feeling is that some powerful category theoretic tools will be necessary to make the subject easier to attack and conceptually more transparent. Such tools have been developed with J.-M. Cordier and they would seem to have many interesting applications outside the immediate area for which they have been developed,(ref. paper [76]).