2-dimensional polyhedra with infinitely many left neighbors

Abstract

In some previous paper D. Kołodziejczyk (2001) [19] we proved that there exists a finite polyhedron P with infinitely many left neighbors in the homotopy category, i.e. P homotopy dominates infinitely many finite polyhedra Pi of different homotopy types and there isnʼt any homotopy type between P and Pi. This answers a question of K. Borsuk (Borsuk, 1975 [2]). The dimension of that P is 3. Here we show that there exists a finite 2-dimensional polyhedron with the same property. Some remarks on constructing examples of 2-dimensional finite polyhedra dominating infinitely many different homotopy types are also included.