A Combinatorial Identity Arising from Cobordism Theory

Let α = (α1, α2, · · · , αm) ∈ R>0. Let αi,j be the vector obtained from α by deleting the entries αi and αj . Besser and Moree [1] introduced some invariants and near invariants related to the solutions ∈ {±1}m−2 of the linear inequality |αi −αj | < 〈 , αi,j〉 < αi +αj , where 〈, 〉 denotes the usual inner product and αi,j the vector obtained from α by deleting αi and αj . The main result of Besser and Moree [1] is extended here to a much more general setting, namely that of certain maps from finite sets to {−1, 1}.