What are some recommended texts for studying algebraic topology?

[Math Amateur]

I have a basic (very basic understanding of the elements of algebra and many years ago I did a course in analysis ... and I would very much like to read my way to an understanding of algebraic topology ..

I figured I should start with some basic texts on topology that (hopefully) head toward a few chapters later in the book on algebraic topology after doing the necessary point-set topology.

I chose the following text for the task:

Essential Topology by Martin D Crossley

I would like to know what MHB members think of this book and ... more importantly, what texts they feel are worth reading/studying if one is embarking on a path toward algebraic topology ...

Peter

 

[Chris L T521]

I have a basic (very basic understanding of the elements of algebra and many years ago I did a course in analysis ... and I would very much like to read my way to an understanding of algebraic topology ..

I figured I should start with some basic texts on topology that (hopefully) head toward a few chapters later in the book on algebraic topology after doing the necessary point-set topology.

I chose the following text for the task:

Essential Topology by Martin D Crossley

I would like to know what MHB members think of this book and ... more importantly, what texts they feel are worth reading/studying if one is embarking on a path toward algebraic topology ...

Peter

From my experience, in addition to algebra, you should have a good foundation in analysis (primarily topology), and be familiar with manifolds/surfaces/topological spaces like $\Bbb{R}^n$, $\Bbb{C}^n$, $\Bbb{RP}^n$, $\Bbb{CP}^n$, $\Bbb{S}^n$, $\Bbb{T}^n$, etc. and at least know what Lie groups are.

Also, Munkres' Topology is the book I learned topology from, and in it is a part (consisting of 6 chapters) that is dedicated to algebraic topology, so I'd suggest looking into that as well for a source.

At the beginning of the subject, you'll be going through a lot of stuff dealing with homotopy groups $\pi_n(S)$ where $S$ is the space of interest (the Fundamental Group is the more well known homotopy group $\pi_1 (S)$). When I took the class, we also did some stuff on surface theory and then used ideas from there to compute fundamental groups of various combinations of surfaces. After than, you'll have tons of fun with homology and co-homology (where Poincaré duality is the important theorem that ties the two together); CW, cellular, simplicial, and $\Delta$ complexes, and other fun things as well.

Once you get to the point of going through Algebraic Topology, I'd recommend Hatcher's Algebraic Topology because I found it to be a pretty decent text, and the best part about it is that the online version is free! (Cool)

I hope this gives you some insight on what to look forward to on your journey. I really want to relearn this stuff myself but I don't have the time right now... ;_;(Smile)

 

[ModusPonens]

From my experience, in addition to algebra, you should have a good foundation in analysis (primarily topology), and be familiar with manifolds/surfaces/topological spaces like $\Bbb{R}^n$, $\Bbb{C}^n$, $\Bbb{RP}^n$, $\Bbb{CP}^n$, $\Bbb{S}^n$, $\Bbb{T}^n$, etc. and at least know what Lie groups are.

Also, Munkres' Topology is the book I learned topology from, and in it is a part (consisting of 6 chapters) that is dedicated to algebraic topology, so I'd suggest looking into that as well for a source.

At the beginning of the subject, you'll be going through a lot of stuff dealing with homotopy groups $\pi_n(S)$ where $S$ is the space of interest (the Fundamental Group is the more well known homotopy group $\pi_1 (S)$). When I took the class, we also did some stuff on surface theory and then used ideas from there to compute fundamental groups of various combinations of surfaces. After than, you'll have tons of fun with homology and co-homology (where Poincaré duality is the important theorem that ties the two together); CW, cellular, simplicial, and $\Delta$ complexes, and other fun things as well.

Once you get to the point of going through Algebraic Topology, I'd recommend Hatcher's Algebraic Topology because I found it to be a pretty decent text, and the best part about it is that the online version is free! (Cool)

I hope this gives you some insight on what to look forward to on your journey. I really want to relearn this stuff myself but I don't have the time right now... ;_;(Smile)

That's the advice I would give regarding topology.

Regarding abstract algebra, you will have to know groups, group actions, rings and modules. I think it's not necessary to have a deep knowledge of these, because the course is about topology, not algebra. Nevertheless, you have to know these things reasonably well. And (abstract) algebra is a very, very beautiful subject, imo, so it won't be wasted time.

This material is contained in Hungerford's "Algebra" in chapters I, II, III and IV. But Hungerford is not a good place to start from scratch _ at all. So, maybe someone can recommend an easier going algebra introduction that contain these subjects.

 

[Math Amateur]

From my experience, in addition to algebra, you should have a good foundation in analysis (primarily topology), and be familiar with manifolds/surfaces/topological spaces like $\Bbb{R}^n$, $\Bbb{C}^n$, $\Bbb{RP}^n$, $\Bbb{CP}^n$, $\Bbb{S}^n$, $\Bbb{T}^n$, etc. and at least know what Lie groups are.

Also, Munkres' Topology is the book I learned topology from, and in it is a part (consisting of 6 chapters) that is dedicated to algebraic topology, so I'd suggest looking into that as well for a source.

At the beginning of the subject, you'll be going through a lot of stuff dealing with homotopy groups $\pi_n(S)$ where $S$ is the space of interest (the Fundamental Group is the more well known homotopy group $\pi_1 (S)$). When I took the class, we also did some stuff on surface theory and then used ideas from there to compute fundamental groups of various combinations of surfaces. After than, you'll have tons of fun with homology and co-homology (where Poincaré duality is the important theorem that ties the two together); CW, cellular, simplicial, and $\Delta$ complexes, and other fun things as well.

Once you get to the point of going through Algebraic Topology, I'd recommend Hatcher's Algebraic Topology because I found it to be a pretty decent text, and the best part about it is that the online version is free! (Cool)

I hope this gives you some insight on what to look forward to on your journey. I really want to relearn this stuff myself but I don't have the time right now... ;_;(Smile)

Thanks Chris interesting picture you paint ... Will definitely spend some time studying the Munkres text

Peter

- - - Updated - - -

That's the advice I would give regarding topology.

Regarding abstract algebra, you will have to know groups, group actions, rings and modules. I think it's not necessary to have a deep knowledge of these, because the course is about topology, not algebra. Nevertheless, you have to know these things reasonably well. And (abstract) algebra is a very, very beautiful subject, imo, so it won't be wasted time.

This material is contained in Hungerford's "Algebra" in chapters I, II, III and IV. But Hungerford is not a good place to start from scratch _ at all. So, maybe someone can recommend an easier going algebra introduction that contain these subjects.

Thanks ModusPonens ... Agree with your opinion on abstract algebra ... Will look up the Hungerford text ...

Peter

 

[blue_raver22]

,

Thank you for your question. I highly recommend the text "Algebraic Topology" by Allen Hatcher. This text is highly regarded in the mathematical community and provides a comprehensive introduction to the subject. It covers both point-set topology and algebraic topology, making it a great starting point for someone with a basic understanding of algebra.

Another recommended text is "Topology" by James R. Munkres. This text is also widely used and covers both point-set topology and algebraic topology in a clear and concise manner. It also includes many examples and exercises to help solidify your understanding of the material.

For a more advanced text, I suggest "Algebraic Topology" by Edwin H. Spanier. This text is more rigorous and goes deeper into the theory behind algebraic topology. It may be more challenging, but it is a valuable resource for those looking to delve deeper into the subject.

Overall, I believe the text you have chosen, "Essential Topology" by Martin D Crossley, is a good starting point for your journey towards algebraic topology. However, I would also recommend supplementing your studies with one or more of the texts mentioned above. Happy reading!