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I am about to give a seminar presentation in an algebra seminar (undergraduate level) about the Hopf fibration $SU(2) \longrightarrow SO(3)$ for which I will use an algebraic-topological approach using the more topological representations $SU(2) \cong S^3$ as well as $SO(3) \cong \mathbb{R}P^3$. For this, I want to introduce the concepts of universal coverings and fundamental groups.

My question now is, how can I give some motivation for math-undergraduates (especially those possible 'only' interested in Algebra) to learn about fundamental groups/universal coverings? For me, it's just an interesting concept, but that's not very convincing, I suppose...
Hence:

What are good reasons to learn about fundamental groups (and universal coverings?) when you're an undergraduate, (possibly mainly) interested in Algebra and not necessarily interested in algebraic topology?

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  • $\begingroup$ Is your question limited to Hopf bifurcations and such, or instead why should undergraduates be interested in group theory overall as a discipline, even if their primary interest may be algebra? $\endgroup$ Jun 7, 2017 at 20:42
  • $\begingroup$ I believe algebra is mainly abstract structure and tools, geometry and topology realizes these in concrete examples(like yours above) $\endgroup$
    – Rust Z
    Jun 7, 2017 at 20:44
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    $\begingroup$ You could mention a couple applications of algebraic topology to algebra, e.g. proving the fundamental theorem of algebra, or using $\pi_1$ to show that a subgroup of a free group is free. $\endgroup$
    – Michael M
    Jun 7, 2017 at 20:45
  • $\begingroup$ @MichaelM Subgroups of free groups are free through usage of $\pi_1$ - yes, I think I had that, good idea! The proof of the fundamental theorem doesn't use $\pi_1$ though, does it (I think we might be thinking of the same one, which is more point-set-topological)$? $\endgroup$ Jun 7, 2017 at 21:00
  • $\begingroup$ I think historically the reasons for studying fundamental groups and homology came from multivariable calculus and differential equations. A closed differential $1$-form is locally exact, but is only globally exact if it is defined on a simply-connected region. If this region is not simply-connected, you can lift the form to the universal cover (which is simply-connected). This spawned the theory of Riemann surfaces, covering spaces, and monodromy. $\endgroup$ Jun 7, 2017 at 21:34

2 Answers 2

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One motivation for an algebra seminar is to illustrate that there are good reasons to learn about groups. Many students may think that groups are just an abstract structure. However, groups arise at many other interesting places, like symmetry groups in geometry and physics, Galois groups in field theory and number theory, fundamental groups in topology and geometry, permutation groups in combinatorics and representation theory, and many other areas.
There are also specific good reasons to learn about fundamental groups. For example, certain compact manifolds are classified by their fundamental groups, namely compact Riemannian-flat manifolds. So the fundamental group "says everything" here.

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  • $\begingroup$ That is a good point I hadn't thought of before... can you maybe refer me to a (the?) 'famous' theorem that states this classification of compact Riemannian-flat manifolds? $\endgroup$ Jun 7, 2017 at 21:02
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    $\begingroup$ There is the book of Charlap, and the book "Spaces of Constant Curvature" by Joe Wolf. Both are excellent books on these subjects. For a paper, see also here, Theorem 5. $\endgroup$ Jun 8, 2017 at 18:14
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From @DietrichBurde's answer, I'm assuming (too) that the original question is about the value of group theory to those interested in algebra, and indeed, the applications of group theory in mathematics he states are excellent.

My own approach to teaching math—and especially physics—at the undergraduate level is a bit different. I try to start with something the students already know, are already interested in, then show them what they don't know about this topic, and show how the discipline before them will empower them to answer those questions. In short, I try to exploit the students' innate curiosity more than some "career" usefulness.

As background: In my introductory optics classes I project a photograph of a beautiful double rainbow, which students all love and think they undertand and ask:

  • "How does a rainbow arise?"
  • "Why this order of colors (red on outside, blue on inside)?"
  • "Why can't you ever get close to a rainbow?"
  • "Why is a rainbow always this size, never bigger, never smaller?"
  • "Why is the larger (secondary) rainbow farther out?"
  • "Why are the colors in the secondary in the opposite order?"
  • "Why is the width of the band in the secondary rainbow larger than in a primary one?"
  • "Why is the broad range of sky between the rainbows ('Alexander's dark band') darker than the rest of the sky?"
  • "Why does the rainbow move left or right when you do?"
  • "What causes the pastel-colored rings just to the inside of the rainbow?"

And I do not answer these questions on the first class. I say, instead, "In this class we'll study optics so you can answer all these questions. Read Chapter 1 of your text. See you Monday."

So for group theory, I'd come to class with a Rubik's cube, and ask the following questions:

  • "How many different configurations?"
  • "Notice I can rotate the cube and the configuration is (effectively) unchanged. How many ways can I rotate and get an 'unchanged' cube? (We call these symmetries.)"
  • "What is the 'most complicated' scrambling of the cube?
  • "What scrambled state takes the most steps to 'fix'?"
  • "If I perform this rotation (A) then that rotation (B), we get a different result from first performing B then A. Why?"

Then I would, wordless, merely hold up a $4 \times 4 \times 4$ Rubik's cube. Then a $5 \times 5 \times 5$ cube.

And I would not answer any of these questions (in the first class).

I'd end the class with: "The branch of mathematics that will enable you to answer these questions is group theory. Read Chapter 1 of your text. See you Monday."

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  • $\begingroup$ Thank you for this answer, however I don't think this approach will work... I'll mainly be dealing with other Math undergraduates who might not care at all for these examples (I know I'm not the types, for instance). I also don't think it all boils down to the importance of group theory alone; rather about what could be fascinating about the fundamental group/what applications in Algebra might it have/ or how this concept would be useful later on when learning other algebraical concepts etc. $\endgroup$ Jun 7, 2017 at 21:23

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