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Minimum numbers decide e.g. whether a given map f: S^m --> S^n/G from asphere into a spherical space form can be deformed to a map f' such that f(x)not equal f'(x) for all x in S^m. In this paper we compare minimum numbers to(geometrically defined) Nielsen numbers (which are more computable). In thestable dimension range these numbers coincide. But already in the firstnonstable range (when m=2n-2) the Kervaire invariant appears as a decisiveadditional obstruction which detects interesting geometric coincidencephenomena. Similar results (involving e.g. Hopf invariants, taken mod 4) areobtained in the next seven dimension ranges (when 1<m-2n+3<9). Theselfcoincidence context yields also a precise geometric criterion for the openquestion whether the Kervaire invariant vanishes on the 126-stem or not.

Kervaire invariants and selfcoincidences