There exists a polyhedron with infinitely many left neighbors

Abstract

We show that there exists a finite polyhedron P homotopy dominating infinitely many finite polyhedra Ki of different homotopy types such that there isn't any homotopy type between P and Ki. This answers negatively the question raised by K. Borsuk in 1975: Does every FANR have only finitely many left neighbors? We assume the reader to be familiar with the basic notions of shape theory. The classical works here are [B2], [DS], [MS]. Following K. Borsuk ([B2], p. 349) we call a shape Sh(X) a left neighbor of Sh(Y) iff X is shape dominated by Y, Y is not shape dominated by X, and every Z such that X is shape dominated by Z and Z is shape dominated by Y has the shape of either X or Y (in the same manner we define a right neighbor of Sh(Y)). Concerning this notion K. Borsuk asked in his monograph in 1975 ([B2], Problem (6.4), p. 349): Is it true that every FANR has only a finite number of left neighbors? We answer this question showing that there exists even a finite polyhedron with infinitely many left neighbors which are also finite polyhedra. Remark 1. Similarly, in the homotopy category of compact ANR's there were defined the so-called h-neighbors (see [B1]). Since on this category shape theory and homotopy theory coincide, we immediately obtain an example of a finite polyhedron with infinitely many left h-neighbors. Let us note that from the results of Hastings and Heller ([HaHel], [HaHe2]), there is a 1-1 correspondence between the shapes of compacta and the homotopy types of CW-complexes dominated by a given polyhedron. Hence we will work in the homotopy category of CW-complexes and homotopy classes of maps between them. Remark 2. On the other side, it is easy to find a finite polyhedron with infinitely many right neighbors. It suffices to take a point {p}, then the spheres Si, for Received by the editors February 28, 1999. 2000 Mathematics Subject Classification. Primary 55P55, 55P15.