Second homotopy classes associated with non-cancellative monoids

We construct second homotopy classes associated with twins of non-cancellative tuples of a monoid, where the monoid is defined by the semi-positive fundamental relations of the fundamental group of a CW-complex. As an application, we reconstruct the second homotopy classes for the complement of generic lines arrangement studied by Akio Hattori. We aim to apply the theory for the complement of elliptic discriminant loci in a forthcoming work.