What's the difference between differential topology and algebraic topology?

Main Question or Discussion Point

Having some knowledge of differential geometry, I want to self-study topology. Which of the two areas shall I study first? Thanks for answer!

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Office_Shredder
Staff Emeritus
Gold Member

Differential topology is the study of manifolds: You consider something that locally looks like Euclidean space, on which you can differentiate etc. The proofs used in differential topology look similar to analysis; lots of epsilons and approximations etc.

Algebraic topology considers a broader class of topological spaces, and assigns algebraic objects (groups) to them. You have lots of algebraic theorems about homomorphisms between groups related to continuous maps between topological spaces and other things like that in order to draw conclusions.

To study differential geometry, you might need some (very) basic topology. Aside from this, they're not very related, as far as I know.

Given a smooth manifold, the two are very much related, in that you can use differential or algebraic techniques to study the topology. Here's an example: prove that the homotopy groups

$$\pi_i(S^n)=0, i<n$$

This is a question in algebraic topology, but by far the simplest proof uses Sard's Theorem, a theorem in differential topology.

I think Algebraic Topology is more general, in that your spaces do not need to admit
a smooth structure for the results of AT to apply.

One theorem relating the two areas is also deRham's theorem.

lavinia
Gold Member
Having some knowledge of differential geometry, I want to self-study topology. Which of the two areas shall I study first? Thanks for answer!
many of the results in differential topology involve differentiable versions of results in algebraic topology. Without knowing the algebraic topology you will miss this connection.

On the other hand beginning differential topology is elementary - largely based on Sard's Theorem and the Implicit Function Theorem - and can be learned as an extension of multivariate calculus.

Learn both at the same time.

Differential topology is the study of smooth manifolds and smooth mappings using only the methods of calculus. Most importantly this means that it avoids the use of metrics - though not completely - and thus is distinguished from Riemannian geometry. When we first learn multi-variate calculus we implicitly use the metric on Euclidean space and are no told that this metric is extra structure that is not really needed. For instance, the idea of the gradient of a function uses the metric. But one could just as well use the differential of a function and discard the metric. This is what happens in differential topology.

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