FOlDING Of CHAIN COMPLEXES BY USING HOMOLOGY GROUPS

Abstract

The project of this thesis concerns a field of mathematics called Geometric Topology which essentially studies various structures and properties of manifolqs.

A continuous map f: KL from a regular CW-complex K

to another L is a cellular folding iff it maps i-cells of K to i-cells of L such that the closures of any cell e and its image f(e) must contain n-distinct vertices. The set of singularities of f makes a cellular subdivision of K, in the case K is a surface then this set has the structure of a graph embedded in K.

The set ofCW-complexes together with cellular folding form a category whose objects are CW-complexes and the morphism are cellular foldings. The cellular folding will be called neat if Ln - Ln-I consists of a single n-cell, 11 Int L 11 • The connected sum of two cellular foldings f,g of surfaces K,

L, where f: K--tP , g: L--tP and P is ann-sided polygon, n n n

we denoted it by h = f# g : K#L--tP and defined it as a cellular n

map such that

e; EK;

e'; ELi ,

f(8ei)=g(8e'i)=h(8e"i), Vi=0,1,2

The thesis consists of three chapters .

Chapter 1: CW-complexes. The category ofCW-complexes is one of the most useful categories of topological spaces. They have sufficiently nice local properties so that the most pathological