Algebraic Structures on a Manifold

In this book we concerned with extending the folding and unfolding of manifolds to a folding and unfolding of a fundamental group. The relations between the folding and the deformation retract on the fundamental group are discussed. The folding and unfolding on a wedge sum and a connected sum of some types of manifolds are obtained. Some types of conditional foldings restricted on the elements of the fundamental groups are deduced. The concept of foldings will be discussed from view point of algebra. The effect of retractions of manifolds on the fundamental group is obtained . Moreover, we introduce new type of the fundamental group called chaotic fundamental group. Some applications in manufacturing are presented.