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Let TCₙ denote the n-th topological complexity. We prove that if X is a CW space of finite type with TCₙ (X) =n-1 then X is homotopy equivalent to an odd-dimensional sphere. Previously this result was known for nilpotent spaces and, in addition, for n=2 and all X (Grant--Lupton--Oprea).
Yuli B. Rudyak (Sun,) studied this question.