On a family of topological invariants similar to homotopy groups

Abstract

The intimacy relation between smooth loops, which is a strong homotopy relation, is generalized to smooth maps defined on the n-cube, leading to a family of groups similar to the classical homotopy groups. The formal resemblance between the two families of groups is explored. Special attention is devoted to the role of these groups as topological invariants for manifolds and as tools for describing geometrical structures defined on manifolds such as bundles and connections.