  Let TC$_n$ denote the n-th topological complexity. For nilpotent CW space of
finite type, Grant--Lupton--Oprea proved the following: If TC$_n$(X)=n-1 then X
is homotopy equivalent to an odd-dimensional sphere. In my previous version I
stated that I can get rid of the nilpotence condition. Currently I cannot do it
(the proof of Theorem 4.4 in the previous version is wrong) but I have made
some progress there, and I present it here
