As an undergraduate student who has studied some point set topology and abstract algebra, I aim to start studying differential topology using Guillemin-Pollack and algebraic topology using Hatcher on my own.

Before this, I want to understand the fundamental differences/similarities between these two subjects.

Is it true that algebraic topology and differential topology are similar subjects in the sense that they seek to solve same kind of problems but using different tools? Or are there problems that can only be solved by one technique? Or in what respects are these two subjects different from one another?

It would also be very helpful if you could direct me to some books which highlight these issues in general.

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    $\begingroup$ See the Wikipedia page for algebraic topology... it does a good job of explaining the basics of what AT is and the problems it tries to answer. It explains that AT and DT can work in the same setting (manifolds) but that each tend to focus on different aspects of a manifold; namely, that AT will attend to global, non-differentiable results and DT will focus on smooth (differentiable) manifolds that give rise to a geometric structure, which we can take advantage of via invariants. Note that this invariance-business is common to both AT and DT since both try to classify things. $\endgroup$ – coreyman317 Oct 25 '18 at 3:03
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    $\begingroup$ Unrelated to your question, but I recommend any text but GP for a first study in differential topology. If you're going to learn AT at the time, then maybe try Tu's Introduction to Manifolds, as that should lead nicely to Bott and Tu's Differential Forms in Algebraic Topology which combine the topics nicely. Alternatively, I'd also recommend Lee's Intro to Topological Manifolds, and Intro to Smooth Manifolds. I think GP's text is too "non-rigorous" for a first exposition to the subject. $\endgroup$ – Matt Oct 25 '18 at 9:40
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    $\begingroup$ There is nothing non-rigorous in Guillemin and Pollack's book. The only objection I consider reasonable is that it prefers to work with submanifolds of $\Bbb R^N$ instead of abstract manifolds; it does this to get to the actual topology as quickly as possible instead of spending a lot of time setting up formalism. This is easily remedied by reading a few early chapters of Lee. $\endgroup$ – user98602 Oct 25 '18 at 11:57

Of course the two fields have a lot in common but I wouldn't say they deal with the same kind of problems. Sure there are problems you can approach with both algebraic and differential topology, for instance you can prove Brower's fixed point theorem (https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem) with both. The algebraic topology proof uses singular homology while a differential topology proof can use something like approximation by a smooth function and Sard's theorem (for sure you can find these proofs in any book on the subjects). Broadly speaking differential topology will care about differentiable structures (and such) and algebraic topology will deal with more general spaces (CW complexes, for instance).

They also have some tools in common, for instance (co)homology. But you'll probably be thinking of it in different ways. An algebraic topologist is more interested in singular homology (https://en.wikipedia.org/wiki/Singular_homology) and a differential topologist in deRham cohomology (https://en.wikipedia.org/wiki/De_Rham_cohomology) or Morse homology (https://en.wikipedia.org/wiki/Morse_homology).

But let me give two examples of problems that are clearly from one area and not the other. The first is the existence of homeomorphic but not diffeomorphic differential manifolds. This is clearly a topic of differential geometry. The first such example was found by Milnor who showed the existence of exotic structures on $S^7$ i.e. a manifold homeomorphic but not diffeomorphic to $S^7$ (https://en.wikipedia.org/wiki/Exotic_sphere).

On the other hand an algebraic topology problem (or more specifically a homotopy theory one) is the computation of the homotopy groups of spheres (https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres). Sure spheres are also smooth manifolds but now we only care about their topology. And this sort of computations use very algebraic techniques like spectral sequences (https://en.wikipedia.org/wiki/Spectral_sequence) and the "weird" Eilenberg-Maclane spaces (https://en.wikipedia.org/wiki/Eilenberg%E2%80%93MacLane_space).

You can also take a look at arxiv and compare the tags "Algebraic topology" and "Geometric topology" (roughly the same as differential topology, I guess) to get an idea of how different the modern stuff is in both fields.

To wrap it up, I think that the best way to understand these fields and appreciate them is to learn at least the basic ideas in both. About books, I first learned algebraic topology by Hatcher and differential topology by Hirsch and I really liked both of them. Milnor's book "Topology from a differentiable point of view" is also very nice.


The above comments and answer by @Miguel Moreira do a good job of explaining the differences between the fields, but I just thought I'd provide some perspective that I can from having just finished a course on Algebraic Topology this past May. The big disclaimer here is that I have never taken a formal class in Differential Topology, but I can offer what I know about Algebraic Topology.

For my class in AT, we used the book written by William Massey. We pretty much just followed the book and ended the course with a study of these things called "covering spaces". But to answer your question, everything we did in that class involved some type of universal property and from talking to others, I'm not quite sure Differential Topology uses universal property notions as much.

We would define Free Abelian Groups as objects that satisfy this commutative diagram, and the hallmark theorem that is the Seifert-Van Kampen theorem was treated in this manner too. To be perfectly honest, what I remember most about that class is just seeing something like "oh, and this works because it satisfies this commutative diagram" absolutely everywhere, so really I feel jaded because all I associate with the subject of Algebraic Topology are commutative diagrams. I know it might sound silly to say but really everything in that class was very algebraic in nature (hence the name of the subject) and so we played around with fundamental groups a lot. While of course continuous functions were still in play (it is topology after all), we didn't care about differentiation on surfaces or manifolds, or if those manifolds were smooth or not. In fact, the word "smooth" never came up in the subject, and for good reason. Basically, for a first course on the subject, we just defined a few key concepts (manifolds, surfaces, homotopy, fundamental group, free groups, free abelian groups, SVK Theorem, covering spaces, local homeomorphisms) and studied how they could be useful when studying topological objects with an algebraic lens. Again, not sure how helpful this is, but this is just my two cents.


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