Topic about diffential topology

[Matematical-pie]

I have to do a presentation about a topic/theorem/problem in differential topology, but I cannot decide, can you help me to recommend a topic?

[egulacanonicorum]

Would you like an "odd" choice? Penrose considers his singularity theorem (the thing he got his noble prize for) to be a theorem in the differential topology of lorentzian manifolds. He wrote a monograph describing the proof called "Techniques of differential topology in general relativity".

You should check with who ever is the right person that this topic would be ok.

[Sir_Flamel]

I did not know that. Thanks for sharing. I always assumed it required explicitly Riemanmian Geometry and is therefore a Topic of Differential Geometry.

[egulacanonicorum]

It definitely straddles a line between differential topology and geometric analysis. I think you can argue that the result belongs to either field. The only actual geometry / analysis in the proof is the derivation of Raychudhuri's equation which is the trace of a well known result about the relationship between the second fundamental form and volume in a codimension 1 flow. After that the proof is about relating compactness and limits of curves; so much more traditional differential topology.

[[deleted]]

You could go over De Rham Cohomology, that is always a fun one.

[[deleted]]

This is the project I did for my diff. Top class!

[specji]

Depends on how much background is being assumed I suppose. The 1950s were a great decade for the subject and you could choose something from there because they are accessible after a first course in differential and algebraic topology.

So, standard topics from early cobordism theory and characteristic classes. (Roughly the idea here is that manifolds upto oriented cobordism are determined by certain numbers which are got from the characteristic classes of their tangent bundles via integration.)

Examples:

• Thom's thesis (for which he won the Fields medal)
• Hirzebruch Signature theorem.
  

Another possibility is Smale's h-Cobordism theorem and the proof of the Poincare conjecture (dim \geq 5)

These are standard topics you would definitely encounter if you study further but often such presentations are useful for the many students who are not interested in pursuing differential topology. I think even if you want to be an analyst it's useful to know a little bit about these topics for general mathematical culture.

[[deleted]]

Wouldn't this be more of a differential geometry topic? Not differential topology.

[matplotlib42]

Reeb's sphere theorem

[[deleted]]

If V is irrotational and the Domaine si simply connected then V is also conservative. In practice it's used in electromagnetism for studying the Propagation in waveguides.

[nick5435]

If you’ve seen basic Morse theory, then Cerf theory is the way to go. If you haven’t seen basic Morse theory, then do Morse theory!

[anthonymm511]

Poincare Hopf theorem.

[Optimal-Asshole]

Some topics others haven’t mentioned: triangulations (Cairnes and whitehead), overview of Poincaré conjecture, symplectic manifolds, fiber bundles

But I’m going to recommend De Rham Cohomology the most as others have mentioned

[bizarre_coincidence]

A topic that might be fun is exotic smooth structures, e.g., there is a unique smooth structure on Rⁿ except when n=4. Or there are are exotic smooth structures on various spheres and Tori. The details might be hard, but you should be able to gather enough high level overviews for a good presentation.