Reading Notes Of Algebraic Topology

Topology is divided into many directions, for example, generaltopology using set theory as its weapon, combative topology choos-ing combination methods, algebraic topology, of cause, using alge-braic tools, and di?erential topology using di?erential analysis.This note of reading is about algebraic topology, in fact, it isabout homology theory. Algebraic topology takes homology theoryand homotopy theory as its main parts. But there are many kindsof homologies, cell homology, singular homology and the relatedcohomologies, and they have the similar results in many times. Sothere are a nature question: are these homologies same? [E-S] is agreat book. It said that homology is a kind of functions.1. Homology maps each pair of topology spaces (X,A) into a se-quence of Abelian groups H_q(X,A), for each integral q.2. Homology maps function f : (X,A)→(Y,B) to a sequence ofhomomorphisms f_q : H_q(X,A)→Hq(Y,B),for each integral q.3. There is a sequence of boundary maps {(?)_q}, such that (?)_q :H_q(X,A)→H_q(A). and the homology must satisfy seven axiom, they are: identity ax-iom, associative axiom, natural axiom, exact axiom, homotopy ax-iom, excision axiom and dimension axiom.They proved in the category of triangular spaces, if two homolo-gies are satisfied the seven axioms, they have a natural isomorphism.In my reading notes. I have introduced chapter 2, which partsabout calculation of homology groups, and I have given a proof ofthe dependents about homology axioms in chapter 1, and I havesolved a problem by homology groups.