Is orientability a miracle?

Question

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$This question is prompted by a recent highly-upvoted question, Conceptual reason why the sign of a permutation is well-defined? The responses made me realize that my intuition differs from that of many other mathematicians, in a way that I had previously been unaware of.

As a bit of personal background, I recall being taught in middle school that (in not so many words) that there are "24 symmetries of a cube," since you can rotate any face to the bottom (a factor of 6) and then rotate that face in place (a factor of 4). Similarly, for each of the Platonic solids, we can count $4\times 3 = 12$ "symmetries" of a tetrahedron, $8\times 3 = 24$ "symmetries" of an octahedron, and so forth.

One of the various equivalent formulations of the aforementioned MO question is, "What is the conceptual explanation for the existence of the alternating group?" In part because of my background, the answer that came to my mind was, "Because it's the group of symmetries of a simplex." However, people rightly pointed out that to conform to standard usage, this answer should really be phrased, "Because it's the group of orientation-preserving symmetries of a simplex." But once the statement is phrased this way, it suggests that maybe the existence of the alternating group isn't such a "basic" fact after all; maybe it's only the symmetric group whose existence can be taken as basic, and we have to "explain" the existence of a subgroup of index 2 using some kind of algebraic argument.

There's no question that Poonen's version of Cartier's argument is slick and beautiful. Nevertheless, something doesn't quite sit right with me if we call this an "explanation" of the existence of the alternating group. It still seems to me that orientation-preserving rigid motions in Euclidean space are mathematically fundamental, because they are rooted in our physical intuition. Admittedly, in modern mathematics, formally capturing our physical intuition in a direct manner is a cumbersome process; we have to construct the real numbers, and then talk about continuous transformations that preserve the metric. But for me, the complexity of this formal definition does not imply that the underlying concept is complex; rather, it says more about the difficulties involved in formalizing our geometric intuition. It is not hard for me to imagine an alternative universe in which we begin with $\SO(n)$, and think of $\O(n)$ as an extension of $\SO(n)$, much as Spin came before Pin.

But let me now finally come to my question. For those who take the symmetric group as given, and feel the need for an algebraic explanation of a subgroup of index 2, do you feel the same way about other finite subgroups of $\SO(n)$? For example, do you think the existence of the hyperoctahedral group of order $2^n n!$ is "obvious," but the subgroup of index 2 is in need of an algebraic "explanation"? If so, is there an algebraic proof that you feel explains all these "miraculous" subgroups of index 2 in a uniform way? Or is the existence of orientability itself a miraculous fact?

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EDIT: Let me make a few additional comments to nudge the discussion in a more technical and less opinion-based direction.

Suppose I don't buy the "argument from geometric intuition" and I seek a more algebraic (or combinatorial) explanation of the existence of the alternating group. It still seems to me that an argument based on determinants, or the distinction between $\SO(n)$ and $\O(n)$, would be a more "conceptual" route than an argument based on actions on the complete graph. For example, suppose I ask the corresponding question for the hyperoctahedral group ("why is there this subgroup of index 2?"). If we've already decided that the relationship between $\SO(n)$ and $\O(n)$ is the key point, then the same explanation works for both questions. Whereas, the graph-theoretic approach doesn't seem to have the same ability to give a unified explanation for both cases.

Am I mistaken about this? Does Cartier's proof generalize to cover all situations that I might try to explain by appealing to the distinction between $\SO(n)$ and $\O(n)$? Does it perhaps generalize even further to explain index-2 subgroups that don't seem to be explicable in terms of $\SO(n)$?

Answer 1

To me, it is not that hard to imagine an alternate universe where the fact that $|\pi_0(GL_n(\mathbb{R}))| = |\pi_0(O_n(\mathbb{R}))| = 2$ is an unstable fact that holds for small $n$, but not in very high dimensions. In such a universe, determinant would only be defined for small square matrices, and $\bigwedge^n \mathbb{R}^n$ would be $0$-dimensional for large $n$.

By way of analogy, suppose we lived in a two dimensional universe. It would be completely intuitive to us that $\pi_1(SL_2)$ was $\mathbb{Z}$. Elementary school textbooks would say something like "every person, throughout their life, has made a certain number of full turns to the left, and a certain number of full turns to the right, and the difference between these numbers is called the 'winding number'." It would then be extremely surprising when we studied three dimensional geometry (that arcane, counterintuitive subject!) and learned that winding number is only defined modulo $2$.

Answer 2

One interesting phenomenon is that there’s a good notion of symmetric tensor category $\mathrm{Rep}(O_t)$ when $t$ is not an integer, but no good notion of $\mathrm{Rep}(SO_t)$ for generic $t$. (Equivalently, $\mathrm{Rep}(O_t)$ doesn’t have a determinant object.). So that’s a clear sense in which something special and surprising happens to allow SO(n) to exist.

Answer 3

In the spirit of the answer by David Speyer, one could also point out that perhaps the important role of 2 (or bilateral symmetry) in our study of symmetry can be seen as an artefact of living in a world where the "special valuation at $\infty$" dominates our intuition. (Since $\mathbb{Z}/\langle 2\rangle$ occurs as the maximal finite subgroup of $\mathbb{R}^{*}$.)

If we lived in a $p$-adic universe for a specific $p\neq\infty$, then the maximal finite subgroup of the group $\mathbb{Z}_p^{*}$ (the units in $p$-adic integers) may have played a bigger role in our intuition.

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The above does not quite work the way I imagined!

The group generated by rotations (defined as the product of two reflections) always (over a field of characteristic different from 2) generates a subgroup of index 2 in the group generated by reflections. My answer to the earlier question mentioned above gives an elementary proof.

So, in this context, the homomorphism to $\mathbb{Z}/\langle 2\rangle$ does appear to play a special role.

Another way to see is by appealing to the Cartan-Dieudonné theorem and determinants.

Answer 4

(Just a long comment that doesn't directly answer the bold question, but does [I hope] address an implicit question.) What I've found fascinating in thinking about your answer from that other thread are the differences between intuition and proof.

Some of the benefits of intuition are:

• It gives us a sense of what is right and wrong, without the need for long complicated proofs. This can significantly shorten the time needed for learning, or the time needed when looking for new ideas, and proofs.
• Similarly, intuition can also tell us when something surprising is happening. It helps us know when to double-check that a proof didn't take a wrong turn.
• It can help us develop a simpler model of the world, and a way to explain known facts to others.

Some of the weaknesses of intuition are:

• It often doesn't give us the greater information included in a full proof.
• It might rely on a single model of a situation, rather than multiple models.
• Intuition might be "just wrong". (Though, those intuitions guided by proofs are often more robust.)
• Even when intuition is right, it is not always possible to convince others that it is helpful/useful/right without a proof.

We've all gone through the process of refining our intuitions. I thought David's example of the 2D beings learning about winding numbers in higher dimensions was perfectly apropos, because I can think of similar "aha!" moments where my intuition had to change. This other question about counterexamples in algebra contains quite a few of them.

Now, the fact that our intuitions differ is a good thing! It means we look at problems in different ways, which can help guide us to new, surprising solutions. Thus, just because I may not (currently) find "orientation-preserving" as intuitive as you, that's not a flaw in either of us. Moreover, the existence of algebraic proofs (whether natural or uniform or not) in this case tell me that your intuition (whatever it is) is a good one!

In the case of the ultrafinitists who question the consistency of PA, we'll really only ever know if their intuition is correct if they eventually find a provable flaw in the fabric of mathematics. My intuition says they won't find one. They, at least, have a hope of proving their intuition is correct.