Reference Request/Independent Study Algebraic Topology

Question

Over my winter break, I am planning on trying to scratch the surface of algebraic topology. I am already comfortable with introductory/intermediate abstract algebra, and topology to the extent of metric spaces.

Are there books which will let me jump right in to Algebraic Topology, or should I pursue general topology first?

If one answer or the other, which books should I use?

I have heard good things about Munkres Topology for general topology, and already have a PDF of Hatcher's Algebraic Topology (although I might prefer to invest in a hard copy to annotate).

Thanks!

Answer 1

It will be better if you don't jump in to algebraic topology straight away. There might be a few concepts you might need to grasp.

For General topology it depends a lot upon your choice. If you want to get to the point and definitions, I would really suggest Munkres . But (this is my preference actually) if you want a geometric overview , a more simple and not so formal approach Armstrong's General topology is very good. Also it has its latter chapters dedicated to algebraic topology . So you can cherish both.

Answer 2

In my undergraduate topology course I remember Jänich's book helped me gain some perspective on the lecture material, but I haven't revisited it since (and I've never tried to read Munkres).

Bredon's Geometry and Topology offers a concise but thorough run-through of general topology beginning with metric spaces. It's everything you need to start Hatcher (or to continue with Bredon). Definitely see if the pace and level work for you if you can find it at the library. Alternatively, here are some notes you could try at a gentler level- https://www.dpmms.cam.ac.uk/~twk/Top.pdf

I often find that Hatcher can be a little too heavy on detail for a first exposure, so it helps to have a good set of notes to guide your way through it. I'm using these- http://tartarus.org/gareth/maths/notes/ii/Algebraic_Topology_2011.pdf