SOME GEOMETRIC TRANSFORMATIONS ON MANIFOLDS AND THEIR ALGEBRAIC STRUCTURES

Abstract

One of the mam techniques of algebraic topology is to study topological spaces by forming algebraic images of them. Most often these algebraic images are groups, but more elaborate structures such as rings, modules; and algebras also arise. The mechanisms which create these images the 'lanterns' of algebraic topology, one might say are known formally as functors and have the characteristic feature that they form images not only of spaces but also of maps. Thus, continuous maps between spaces are projected onto homomorphisms between their algebraic images, so topologically related spaces have algebraically related images. To show that two spaces are not homeomorphic involves showing that such a map does not exist. To do this is often harder as an algebraic means to tackle such problem , Bet associated with each topological space a certain sequence of a group called its fundamental group . As the title of this thesis suggests we are concerned with extending the folding and unfolding of manifolds to a folding and unfolding of a fundamental groups. The relations between the folding , retraction and the deformation retract on the fundamental group are discussed . Some types of conditional foldings and unfoldings restricted on the elements of the fundamental groups are deduced.

The thesis consists of five chapters: Chapter One is an introduction and presents a brief survey of the main important definitions that help us to follow up this work.

In Chapter Two we introduce the results of some geometric transformations of the manifold on the fundamental group. Some types of deformation retracts of the manifold will be discussed. The chain of foldings and chain of unfoldings will deduce a chains of fundamental groups . We introduce the concept of unfolding on the fundamental group. The folding on a wedge sum of some types of manifolds which are determined by