I am now considering about studying algebraic topology. There are a lot of books about it, and I want to choose the most comprehensive book among them.

I have a solid background in Abstract Algebra, and also have knowledge on Homological Algebra(in fact I am now study Tor and Ext functors). But my knowledge in topology is poor, very poor. I could only remember the concepts, but I got no idea with problems.

Please feel freely suggesting me some book to work on.

Thank for reading.

**Answer**

If you would like to learn algebraic topology very well, then I think that you will need to learn some point-set topology. I would recommend you to read chapters 2-3 of *Topology: A First Course* by James Munkres for the elements of point-set topology. If you would like to learn algebraic topology as soon as possible, then you should perhaps read this text selectively. In particular, I would recommend you to focus mainly on the following (fundamental) notions, reading more if time permits:

- Topological space
- Basis for a topology
- The product topology (on a finite cartesian product; if time permits, you can read about the case of an infinite cartesian product but this is not urgently needed as far as algebraic topology is concerned)
- Subspace topology
- Closed set and limit point
- Continuous function
- Metric space
- Quotient topology (this is a very important, but sometimes ignored, prerequisite for algebraic topology)
- Connected space
- Component and path component
- Compact space
- Hausdorff space
- The separation axioms, Urysohn’s lemma and the Tietze extension theorem (if time permits; these are very useful and inteteresting concepts but you can take the Urysohn lemma and Tietze extension theorem on faith if you desire)

I think that as far as algebraic topology is concerned, there are two options that I would recommend: *Elements of Algebraic Topology* by James Munkres or chapter 8 onwards of *Topology: A First Course* by James Munkres. The latter reference is very good if you wish to learn more about the fundamental group. However, the former reference is nearly 450 pages in length and provides a fairly detailed account of homology and cohomology. I really enjoyed reading *Elements of Algebraic Topology* by James Munkres and would highly recommend it. In particular, I think a good plan would be:

- Learn the elements of point-set topology as outlined above.
- Read chapter 8 of Munkres’
*Topology: A First Course*to learn the rudiments of the fundamental group. - Read
*Elements of Algebraic Topology*by James Munkres.

You will not need to know anything about manifolds to read *Elements of Algebraic Topology* but I believe that it is good to at least concurrently learn about them as you learn algebraic topology; the two subjects complement each other very well. I think a very good textbook for the theory of differentiable manifolds is *An Introduction to Differentiable Manifolds and Riemannian Geometry* by William Boothby (but this is a matter of personal taste; there are (obviously) many other excellent textbooks on this subject). The advantage of this textbook from the point of view of this question is that there is a flavor of algebraic topology present in one of the chapters.

I hope this helps!

**Attribution***Source : Link , Question Author : Arsenaler , Answer Author : Amitesh Datta*