Algebraic topology

I am a graduate student in algebraic topology. My family knows nothing about mathematics, and as far as I know can barely solve a linear equation. They sometimes ask me what I’m working on and I usually struggle to explain it. I can throw around some buzz words like it’s the study of continuous deformation or a more fundemental study of shapes and spaces, but I have a hard time explaining my field in simple and understandable, yet exicting way for a layperson. So I guess that I’m asking is if any of you have a good pop-science style articles or even videoes about algebraic topology? The less mathematical the better.

I'm self learning math, a few months ago I finished Lee's topology book and I think I want to learn more about algebraic topology and related category theory topics. I found a few book recommendations online for algebraic topology:

• Hatcher - Algebraic Topology
  this seems to be the most popular one as well as the most hated one

• Dieck - Algebraic Topology
  just looking at it, it seems very long and dense, but also looks reasonable to learn from

• May - A Concise Course in Algebraic Topology
  as the name suggests, this might be hard to learn from, but also seems to cover a lot of topics

• Aguilar, Gitler, Prieto - Algebraic Topology from a Homotopical Viewpoint
  it looks like it omits a lot of the homological algebra stuff but covers topics like vector bundles and k-theory

(There are a few more I found like Massey or Rotman but they seem "too introductory", as in much of their content is about concepts I already learned about from the topology book by Lee, I might be wrong in dismissing these though)

Right now I'm thinking of learning from the one by Aguilar but I'm not sure if I'm going to miss out on too much from the omitted homology topics and how could I make up for that.

I also recently spent some of my time learning category theory from various sources, I understand the basic concepts as well as adjunctions and monads (to some degree), not much more than that. I'm not sure where and what should I read about further in this topic since this seems to be a very broad field but I guess it'll be more clear once I learn more algebraic topology.

Any help on what directions should I take learning these topics would be nice

PhD student here. I am required to take one additional course next semester and have narrowed my choices down to an algebraic topology course or a PDE course.

The only AT I have done is towards the end of my graduate topology course where we got to around the Van Kampen theorem but I don't remember much of it aside from the basics such as what a fundamental group is so I guess you can say I don't know much.

As for PDEs, I have some experience with them from undergrad where we followed Strauss' text, but I thought it might be nice to get a taste of rigorous PDE (I believe we would be using Evan's text).

I'm torn between these two courses and I was wondering if anyone could sway me to take one over the other.

So I've been reading about the Galois theory of covering maps and been staring at this equation for way too long:

$$\left| \pi_1(X,x_0)/p_*(\pi_1(\tilde{X},\tilde{x}_0))\right| = |p^{-1}(x_0)|$$

(And the subgroup is normal iff the covering map is Galois and so the quotient is isomorphic to the set of deck transformations which is exactly singularly transitive on the fibers etc)

So [;p;], the covering map is famously surjective, famously not at all injective. And the induced homomorphism [;p_*;] is the opposite: Injective, (by homotopy lifting) but not at all surjective. And this equation says that the degree that [;p_*;] fails to be surjective (the cokernel) is exactly the same as the degree [;p;] fails to be injective, at least at [;x_0;]. Is there a dual to this notion? Some sort of type of injective continuous map (so an embedding [;\iota;]) that always induces a surjection [;\iota_*;], so that the [;\text{ker}(\iota_*);] is somehow related to how [;\iota;] fails to be surjective? Am I getting anywhere with this?

Some rudimentary work: My first guess is something to do with retractions, since [;\iota_*;] would be easily surjective if there existed a retraction onto [;\iota(U);]. But that's sufficient but not necessary, if you take say [;\iota :S^1 \hookrightarrow S^2;]. In particular if I'm up to something it should be able to distinguish between [;\iota :S^1 \hookrightarrow S^2;] and [;\iota : S^1 \rightarrow S^1 \times [0,1];], one being injective and the other not.

Another way I've thought about it is that the idiots version of a covering map would be a function [;p:X\times D\rightarrow X;] where [;X;] is a discrete space. Every covering map looks like this locally, and the value of the equation up top would be [; D;]. Then the dual would probably look something like [;\iota :X\rightarrow X\times D;]. But then the kernel would be trivial, and not have cardinality [;D;] like it should.

I want to study Algebraic Topology as a way to celebrate my birthday, I wanted to start learning this since long time but never got time, I just graduated from High school, can you please advice me over few resources.

In topology we want to classify spaces up to homeomorphism and by topological invariants. I was told algebraic topology tries to achieve this goal by “mapping” problems in topology to an algebraic setting and using algebraic invariants to classify topological spaces. But what are examples of algebraic invariants and how is topology mapped to algebra?

Any good recommendations for algebraic topology? I know about Munkres and that’s the about the level I’m at, but anything else?

I always see people recommend Munkres’ Topology for introductory topology, but then nobody ever mentions it for algebraic topology, even though it does have a sizable few hundred pages devoted to algebraic topology. Was curious what people’s thoughts on it were.

Thanks.

Next semester I will be taking an AT course, and I wanted to also do a guided reading project based on a related topic. I have already studied some AT on my own and liked it a lot, so I want something of a similar flavor but without much overlap with the course.

What motivated me to take up this project was reading about the and equivalent results like Brouwer's Fixed Point Theorem and Sperner's Lemma. These are the kinds of results I would like to learn about in the project, but I don't know where I can find enough such results to be able to fill up a semester-long project.

My guide has suggested reading 'Geometry of Surfaces' by Stilwell or 'The Knot Book' by Adams, but he is open to anything else I might like. Do you have any suggestions for topics/books I could look into?

Any that you know of?Perhaps in topological QFT?Im interested in the subject but i also dont want to waste time learning it if its not connected to physical applications/theory.

Hey, we're doing an Algebraic Topology reading group starting in a few days. I know groups such as these often die out but we do have a few commited members already ! The goal is to work on homotopy theory and simplicial (and related) homology theory, and manage a VC session each week

If you want to join us, we will be talking on Discord, at

We will be using Bredon "Topology and Geometry", and we're starting at chapter III this week. Prerequisites would be topology, some algebra (group and ring theory is a minimum), and if possible the basics of smooth manifolds (the usual manifolds, notion of a chart etc).

Hope to see you on Discord :)

EDIT: I've gotten a bunch of responses, which is great! I think I've responded to everyone, but if you sent me a chat or a message and I didn't respond, please try again. Everyone who messages me gets the invite, so if you didn't get one, it's just because I'm dumb and didn't see the message. :-)

Hi everyone! My name is Joe and I have a in mathematics where I studied Homotopy Theory and Homological Algebra.

A few friends and I have been meeting once per week for a topology "study group", for lack of a better thing to call it. We work through a text, ask questions, do exercises, and try to support each other as we learn. The exact structure of the meetings changes depending on who is attending, as a lot of people come and go based on content and their personal lives.

We're about to start a new topic, so I thought it would be a good time to invite other people to join us! We're starting Algebraic Topology using (which he gives out for free). We meet for two hours on Sundays at 11am EDT (3pm GMT). We meet over voice chat with some screen-sharing; you're free to join the conversation or just listen if you want. (Sometimes people don't have microphones.) It's a pretty small group but we try not to single anyone out.

Send me a PM if you want to join and I'll get you an invite to the server! Everyone is welcome, I just don't think it's a good idea to put server invites in permanent posts. I'll be checking in on this post and on my PM's throughout the day.

Note: Don't worry too much about "prerequisites". If you have an inquisitive mind and want to come check it out then you should do that. It's not a class, so you can come and go as you please. However, to get the most out of Hatcher's book, you'd ideally have some experience with proof-based mathematics and know some point-set topology.

Hope to see you tomorrow!

I am an undergraduate wondering about the research opportunities for grad school. Are there still many unsolved problems or would I have a hard time finding them?

I just started working through Hatcher's Algebraic Topology and I think it would be really cool if I had some people to talk to about algebraic topology. Maybe we could make a discord server or something?

Looking forward to hearing your thoughts!

Dear all,

I am a graduate student in a Germany university. Today, I took my first ever oral exam on algebraic topology and I failed it. According to university policy, I have two re-takes. I have not decided yet, but I suppose the next one would be around same date next month.

I am in fact an aspiring combinatorialist and theoretical computer scientist, so I’m interested in extremal combinatorics, graph theory, and algorithms. I never felt clicked with topology — I took a general topology course as an undergraduate in an US university, at the end, I think my grade is B or something. (I’m feeling like Szemeredi in Moscow… though I’m no where near as good as he is.)

So my question is: How should I study so that I at least pass this course? With respects for topologists, when studying for the exams, I sometimes just find topology to be really boring and do not want to study... There is a note for this course. We covered topological spaces, basic category theory, fundamental groups and covering spaces, as well as basic abélien group theory.

Thank you for your time!

hopefully not a violation of rules 4 or 5 (I think the homework problem is unsolvable and the math is a bit abstract), but thought people here would enjoy

edit: added the image

Hi everyone. I'm taking undergraduate algebraic topology and algebraic geometry this semester (semester is 3 weeks old). As an international student trapped outside of borders I'm unable to attend in-person tutorials. I'm not very satisfied with the way tutes have gone. I'm looking for people who are also studying the subject, whether it be in a college or independently to discuss with. Anyone interested?

Also, if you have any recommendations on how I can improve my experience, please let me know!

It feels isolating, and also kind of useless! Most of the students are domestic and having in person tutes. It's just me and one other guy noodling around on Zoom.

Hi! I am currently having my second course on Algebraic Topology and it seems that the further we delve into the topic, the more algebraic it gets. I actually quite enjoy Algebra, so I have no problem with it, but I've been wondering, what would you say is the actual algebra/topology ratio that the branch has?