Peter said:I would like to gain an understanding of the basics of Homology theory.
Can MHB members please give some guidance as to how to approach this endeavour.
What do MHB members suggest are good texts or good online resources?
Peter
ThePerfectHacker said:The problem I had with learning algebraic topology is that so many books are hand-wavy. I understand they do that because they want to try to motivate but it seems most of these authors forget that the people learning the subject need to see it done formally too and cannot just wave their hands as simply as the authors do.
A pretty formal book on homology is by Vick. I used it for chapter one and I was pretty satisfied with it. You can read chapter one from that book. But it is sometimes a little concise.
You can also read the last chapter from "Introduction to Topological Manifolds". Lee motivates and carefully does his arguments.
Never use Hatcher. That book is awful.
Homology theory is a branch of mathematics that studies the relationships between different objects and structures by comparing their underlying structures. It is often used to understand the similarities and differences between different shapes, spaces, and other mathematical objects.
Homology theory has various applications in mathematics, physics, and other fields. It is used in topology to classify and study spaces, in algebraic geometry to understand the properties of algebraic varieties, and in biology to study the evolutionary relationships between species.
Homology groups are algebraic structures that are used in homology theory to measure the similarity or difference between mathematical objects. They are defined in terms of cycles and boundaries, and can be used to classify and compare objects based on their underlying structure.
Homology and cohomology are two related but distinct branches of algebraic topology. Homology measures the similarity between spaces based on their underlying structure, whereas cohomology measures the differences between spaces based on their coarser structure. In other words, homology is a measure of the "holes" in a space, while cohomology is a measure of the "cycles" in a space.
Some common tools and techniques used in homology theory include chain complexes, simplicial complexes, homotopy groups, spectral sequences, and category theory. These tools help to define and calculate homology groups, and can also be used to prove theorems and solve problems in homology theory.