Various types of Whitney maps on n-dimensional compact connected polyhedra ( n ⩾ 2)

Abstract

In [6], we defined an adequate index n( P) for 1-dimensional compact connected polyhedron P and we showed that Fd ω −1( t) ⩽ n( P) − 1 for any Whitney map ω for C( P) and any t ∈ [0, ω( P)]. In this paper, we show that the similar result is not always true if P is any n-dimensional compact connected polyhedron ( n ⩾ 2). In fact, we show the following: Let P be any n-dimensional compact connected polyhedron ( n ⩾ 2). Let m be any natural number such that m ⩾ 2. Then there exists a Whitney map ω for C( P) such that the m-sphere S m is homotopically dominated by ω −1( t) for some t ∈ (0, ω( P)). In particular, Fd ω −1( t) ⩾ m.