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It is well known that finite-dimensional Cis-equivalent compacta are shape equivalent (see [D-S]). The firstexample that shows the gap between shape equivalence and CE-equivalence was found by Ferry [Pel]. In [Fe3], it was shown that UV"'-equivalent n-dimensional compacta are shape equivalent. Next Daverman and Venema [D-V] constructed an n-dimensional LC"~2-continuum which is shape equivalent but not (/V^-equivalent to S1. Mrozik [Mrl] obtained a method to have continua which are shape equivalent but not UV1equivalent to each other. Moreover Mrozik [Mr2] improved the method and had a strategy to construct a LC "-continuum Y from any LCn+'-continuum X with tci{X)infinitesuch that they are shape equivalent but not t/F"+1-equivalent. As a criterion of UV "-equivalence he introduced the notions of UV "-component 7i^n\X) [Mrl], yfe-thf/V'-homotopy group nk(n){X) and k-th.C£-homotopy group 7zkCE(X)[Mr2]. Venema [Ve] investigated the groups and showed that 7rAa +1)(Z) = 7tkat+*)(X)=■■■ =KkCE(X) for every continuum X and that 7tnin\Y)=0 for every UV "-continuum Y. In this paper we consider a relation between 7rk(k+1)(X)and the &-th strong shape group zk(X) [Q]. We define a natural homomorphism sk: 7ck(k+1)(X)^ Ek(X) and show that, if pro-TixiX)is profinite,sk is an isomorphism. As its

A relation between $k$-th $UV^{k+1}$ groups and $k$-th strong shape groups