Is there any intro topics involving topology and physics?

[ozone]

I have recently been assigned a project in my undergraduate topology class. I would like to do something in physics which involves topology, but I am having trouble finding a basic topic. I understand that there are some very advanced topics in string theory and the like, but I would like to find something that is more accessible.

I have taken an undergraduate course in differential geometry, and graduate courses in geometrical methods in physics and general relativity. Hopefully there is a good topic available for someone with my limited knowledge. I'm open to any suggestions, but I'd like it to be something I could actually grasp and give a presentation on! Our Prof. recommended these 15 topics for a project. Perhaps one of these is relevant to physics?

(1) Topology of the Cantor set
(2) Topology of Sn and RPn
(3) Topology of simplicial complexes
(4) Topology of algebraic varieties
(5) History of the Euler characteristic
(6) Ham Sandwich theorem
(7) Hairy Ball Theorem
(8) Borsuk-Ulam Theorem
(8) Borsuk-Ulam Theorem
(9) Bolzano-Weierstrauss Property
(10) Covering spaces
(11) Proof of Fundamental Theorem of Algebra using topology
(12) Winding number
(14) Seifert van Kampen and the Fundamental groups of surfaces
(15) Poincare Conjecture

 

[jgens]

Prof. recommended these 15 topics for a project. Perhaps one of these is relevant to physics?

Lots of topics below are actually relevant to physics. Unfortunately physics is not really my area of study, so I can only give a fairly shallow indication of results that are immediately of some interest in physics, but hopefully it can get you started until someone more knowledgeable drops by.

(7) Hairy Ball Theorem

This result essentially just states that there are no non-vanishing vector fields on even-dimensional spheres. An immediate corollary is that connected even-dimensional spheres cannot be Lie groups, which at least seems like it might interest a physicist.

(8) Borsuk-Ulam Theorem

This result, which states every map Sⁿ→Rⁿ takes some pair of antipodal points into the same value, has a very direct connection to physics. In particular, the n = 2 case implies that there are antipodal points on the Earth with the same temperature and barometric pressure.

(14) Seifert van Kampen and the Fundamental groups of surfaces

This is actually a very cool result. Basically you find that closed surfaces are classified by their fundamental groups so there is a complete set of homotopy invariants to distinguish surfaces. The standard homotopy invariants already fail in the 3-dimensional case, where the lens spaces have identical homotopy groups but are non-homeomorphic, so the result is really something special. Not sure how to tie this into physics, but it is pretty interesting nonetheless.

 

[George Jones]

Another possibility is (10) covering spaces. When we move from classical to quantum mechanics, the rotation group SO(3,R) moves to its cover, SU(2), and the restricted Lorentz group moves to its cover, SL(2,C).

 

[dextercioby]

And (12). If the 'winding number' means what I think it means, there's no fully rigorous theory of gauge symmetry without mentioning it.

 

[jgens]

And (12). If the 'winding number' means what I think it means, there's no fully rigorous theory of gauge symmetry without mentioning it.

In this case the winding number is the degree of a map S¹→S¹. It has connections to complex analysis where there is actually a concrete formula computing this number. Not sure if that is what you thought or not.

 

[FactChecker]

Winding numbers are used in complex contour integration for solving electrical potential problems. The signs of the residues depend on the winding being clockwise or counterclockwise. A simpler application is using winding numbers to determine if a point is inside a closed curve. I'm not sure how to tie that directly to physics except for the previously mentioned contour integrals.