Groups of flagged homotopies and higher gauge theory (Q1972810)

The usual homotopy groups \(\pi _{k}(X,p)\) of a manifold \(X\), \(p\in X\) and \( k>1\), are extended in the paper to non-commutative groups \(\Pi _{k}(X;\sigma)\), using ``flagged homotopies'' \(\sigma \) on \(X\), defined in the paper; the groups \(\pi _{k}(X;p)\) are obtained as the limit case, when \(\sigma \) is reduced to the point \(p\). Considering exterior forms with values in a suitable algebra \(A\), the space of cohomologies with coefficients in \(A\) is defined as the moduli space of integrable forms with respect to higher gauge transforms; the action is defined by representations of \(\Pi _{k}(X;\sigma)\) which correspond to integrable forms. Two applications in the cases of the de Rham theory (when \( A=I R\)) and gauge theory are also given.