Conjecture: Minimal higher topological complexity implies odd-sphere homotopy type

Establish that for any connected CW space of finite type X and any integer n ≥ 2, if the nth topological complexity TC^n(X) equals n − 1, then X is homotopy equivalent to an odd-dimensional sphere S^{2r+1} for some r ≥ 0.

Background

Higher topological complexity TC^n(X) is defined as the sectional category of a fibration associated to the diagonal map X → X^n. For non-contractible spaces, the known lower bound TC^n(X) ≥ n − 1 holds, and odd spheres S^{2k+1} satisfy TC^n(S^{2k+1}) = n − 1 for n ≥ 2. This motivates characterizing spaces that achieve this minimal value.

The paper surveys known partial results, including characterization in the simply connected case and improvements under additional hypotheses. The conjecture asserts a complete characterization: minimal higher topological complexity should force X to have the homotopy type of an odd-dimensional sphere.