Adjointness of Suspension and Shape Path Functors

Abstract

In this paper, we introduce a subcategory ∼Sh* of Sh* and obtain some results in this subcategory. First we show that there is a natural bijection Sh(∑(X, x), (Y,y))≅Sh((X,x),Sh((I, Ī),(Y,y))), for every (Y,y)∈ ~Sh* and (X,x)∈ Sh*. By this fact, we prove that for any pointed topological space (X,x) in ∼Sh*,πntop(X,x)≅ πn-ktop(Sh((Sk, *),(X,x)), ex), for all 1≤k ≤n-1.

Keywords

• shape category
• topological shape homotopy group
• shape group
• suspensions