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

In summary, differential topology is the study of manifolds, while algebraic topology is the study of spaces that admit a smooth structure. DeRham's theorem links the two areas. Both areas are important, but beginning with differential topology is elementary and can be learned as an extension of multivariate calculus.
  • #1
petergreat
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4
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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  • #2


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.
 
  • #3


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

[tex]\pi_i(S^n)=0, i<n[/tex]

This is a question in algebraic topology, but by far the simplest proof uses Sard's Theorem, a theorem in differential topology.
 
  • #5


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.
 
  • #6
petergreat said:
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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1. What is the main focus of differential topology?

Differential topology studies smooth manifolds and their properties, such as smooth maps, smooth structures, and smooth transformations. It also explores concepts such as tangent spaces, vector fields, and differential forms.

2. How does algebraic topology differ from differential topology?

Algebraic topology focuses on studying topological spaces through algebraic tools and techniques. It uses algebraic structures such as groups, rings, and modules to understand the global properties of topological spaces, rather than focusing on local properties like differential topology does.

3. What is the role of homotopy in differential topology?

Homotopy is a fundamental concept in differential topology that studies continuous deformations of maps and spaces. It allows for the classification of smooth manifolds up to smooth equivalence, as well as the study of invariants such as the fundamental group and homotopy groups.

4. Can differential topology be applied to real-world problems?

Yes, differential topology has many practical applications, such as in physics, engineering, and computer graphics. For example, smooth manifolds are used to model physical systems, and differential geometry is used to understand the curvature of space-time in general relativity.

5. How does algebraic topology contribute to other areas of mathematics?

Algebraic topology has connections to many other areas of mathematics, such as algebraic geometry, number theory, and mathematical physics. It provides powerful tools for studying and understanding the underlying algebraic structures of these fields, allowing for a deeper understanding of their properties.

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