# Should I learn Algebraic Topology?

• wotanub
In summary, the conversation revolved around the topic of topology and its relevance to physics. The speaker mentioned their interest in learning more about topology and how it relates to their studies in physics. They received recommendations for books and were advised to have some background in real analysis and differential geometry before diving into topology. The conversation also touched on the usefulness of topology in physics and its connection to string theory and general relativity. Overall, the speakers agreed that studying topology is beneficial for developing abstract thinking skills and can have practical applications in physics.
wotanub
I'm a phyiscs student and I have been looking at these lectures:

But I have never learned anything about topology before and was he covers doesn't look like the Topology chapter in my mathematical physics book. I was looking for something maybe for undergrad math majors that goes a little more slowly than my book. This seems to be the pace I want but, while interesting, doesn't seem to be the same topic.

The book starts with what is an open set, neighborhoods, ect but the videos are just talking about shapes. He said this stuff is relevant to physics, but I'm wondering if I'm missing the fundamental stuff? I've just finished studying algebra (in the modern sense).

You have to learn point-set topology before attempting algebraic topology. If you don't already know point-set topology then start with that. Algebraic topology courses will already assume you know it.

A very gentle and more or less popular introduction to topology is Munkres (it essentially holds your hand throughout). At the other end of the spectrum is Willard which is quite terse but has extremely instructive (and infamously difficult) problem sets. You could also check out Bredon which presents a relatively concise introduction to point-set topology and then jumps straight into differential/algebraic topology.

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Traditionally, to study topology, you'd start with real analysis. Once you have done that, point-set topology makes more sense because you know where the abstractions are coming from, and you learn how to do proofs. You can just do point-set, but then you don't see the point (sorry for the very bad pun).

Differential geometry/topology are also good background to have. A lot of the tools of algebraic topology were introduced to study manifolds, which are a concept that originated in differential geometry (from a less abstract viewpoint, they are just generalizations of the kind of surfaces you come across in multi-variable calculus). Algebraic topology can be understood with just point-set, but it's more motivated if you know something manifolds already (in fact, I believe Michael Hutchings, a math professor at Berkeley, said that he doesn't think it makes sense to study algebraic topology without differential topology).

I would recommend at least a little bit of real analysis as suggested. I'm taking a point-set course (with the Moore method) now and I am the weakest student of 4 as it is my first pure math experience since the introduction to proofs class. The course from which this class was cloned had real analysis as a prerequisite as well. I also find it very hard to be motivated in the course, perhaps real analysis would have assisted my motivation in the class.

You'll see concepts like open balls and open neighborhoods around points in differential geometry and even in classical mechanics depending on the text used.

Differential geometry also allows you to use concepts from calculus I-III to build up some other notions about functions from one surface to another. It's quite interesting actually.

As far as finding a book, everyone learns differently from different texts (a fact I am always relearning). Check out your school's library and see if you can use an inter-library loan to find a decent book. Why buy something just to test it out? Let the school/municipality do that and buy one you know you like.

Thanks for the replies guys. I talked to a professor as well and he recommended Munkres and Lee's Topological Manifold book. I also got an analysis recommendation from him. Our library is really good, we have pretty much everything.

I'm going to slowly work thorough them as time permits I guess. None of this is really related to my research, I just think math is interesting as long as it's at least vaguely related to physics.

wotanub said:
None of this is really related to my research, I just think math is interesting as long as it's at least vaguely related to physics.

Then you should definitely check out the following:

https://www.amazon.com/dp/1461426820/?tag=pfamazon01-20

I've been reading it on and off and it's quite amazing. I don't think physics gets more elegant than gauge field theory.

wotanub said:
Thanks for the replies guys. I talked to a professor as well and he recommended Munkres and Lee's Topological Manifold book. I also got an analysis recommendation from him. Our library is really good, we have pretty much everything.

I'm going to slowly work thorough them as time permits I guess. None of this is really related to my research, I just think math is interesting as long as it's at least vaguely related to physics.

Doing it because you enjoy it is definitely a good attitude, since its relationship to physics is actually quite tenuous.

Doing it because you enjoy it is definitely a good attitude, since its relationship to physics is actually quite tenuous.

If you don't consider string theory to be physics, maybe. Roger Penrose used topology in his singularity theorem for black holes, and since then differential topology techniques have become standard part of general relativity. Plus topology and differential geometry have some relation to each other, so the topology helps that way with GR, as well. And physics is pretty big, so it's hard to know what else is out there.

That's the thing about math. Sometimes, there's a gigantic branch of math devoted to something that only has this little teeny-tiny niche application somewhere. So, it gives the feeling of being completely useless, but the teeny-tiny application could be important. It's like rubber. What an obscure material. Some weird thing that comes out of some trees in South America. But somehow, it's pretty important.

I say this as a disillusioned topologist who actually doesn't think topology is that useful, but I think knowing a little basic topology might be a better idea than it could seem to someone who doesn't know much topology. For one thing, it helps your higher-dimensional intuition and ability to think abstractly. And occasionally, stuff like the fundamental group or covering spaces will come up. So, for example, you have SU(2) and SO(3) and SU(2) is a double cover of SO(3). That can help to understand something like spin in quantum mechanics. It's nice to be able to understand the fundamental group of SO(3) and how it relates to this. The belt trick and spinors and all that stuff.

Also, something like anyonic particles. There, you need to know about the braid group, which is the fundamental group of the configuration space of n point particles confined to a plane. And I'm sure there's more.

## 1. Why should I learn Algebraic Topology?

Algebraic Topology is a branch of mathematics that studies the properties of geometric spaces through algebraic tools. It allows us to understand and classify topological spaces, which can be applied in various fields such as physics, computer science, and engineering. Learning Algebraic Topology can also improve your problem-solving and critical thinking skills.

## 2. Is Algebraic Topology difficult to learn?

Like any other branch of mathematics, Algebraic Topology can be challenging to learn. It involves abstract concepts and requires a strong foundation in algebra and topology. However, with dedication and practice, anyone can learn and understand the fundamentals of Algebraic Topology.

## 3. How is Algebraic Topology different from other branches of mathematics?

Algebraic Topology is unique because it studies the invariants of topological spaces, which are preserved under continuous deformations. This allows us to classify spaces and understand their properties without relying on specific coordinates or shapes. In contrast, other branches of mathematics, such as calculus or linear algebra, focus on specific functions or vector spaces.

## 4. How can Algebraic Topology be applied in real life?

Algebraic Topology has various applications in real life, such as in computer graphics, data analysis, and robotics. It also has practical uses in physics, such as in the study of quantum mechanics and general relativity. Additionally, understanding Algebraic Topology can help in problem-solving and critical thinking in various fields.

## 5. Do I need any prerequisites to learn Algebraic Topology?

To learn Algebraic Topology, you should have a strong foundation in linear algebra, abstract algebra, and topology. It would also be helpful to have some knowledge of calculus and analysis. However, with dedication and practice, you can still learn Algebraic Topology without having a deep understanding of these subjects.

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