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Inspired by When is 2 qualitatively different from 3?

Also similar to Are there mathematical concepts that exist in dimension 4, but not in dimension 3? (Math SE), but with the restriction of being regular up to $n=3$.

There are many examples of when 3 is different from 2, but I would have liked to have the follow-up question. What are mathematical features that are more or less regular up to 3 but change dramatically in complexity when the number is increased to 4?

For the moment I can only think of examples of higher orders, like polynomials of order 5 having no general solution. Maybe this question can be extended to the next $n$.

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    $\begingroup$ There are plenty of examples of $n=4$ being weird in topology, e.g. exotic $\mathbb{R}^4$’s. $\endgroup$
    – Sam Hopkins
    Commented Nov 10 at 18:46
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    $\begingroup$ When the $n$-th post of the series will get no answer, we could conclude that $n+1$ is not qualitatively different from $1,\dots, n$, therefore it is qualitatively different. $\endgroup$
    – Pietro Majer
    Commented Nov 10 at 19:47
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    $\begingroup$ I’m voting to close this question because one can keep up this type of question ad infinitum $\endgroup$
    – Benjamin Steinberg
    Commented Nov 10 at 22:05
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    $\begingroup$ I'm voting to close the integers because one can keep up that type of number ad infinitum. $\endgroup$
    – Gerry Myerson
    Commented Nov 10 at 23:51
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    $\begingroup$ In all seriousness, I think I'd actually get something out of perusing more lists like this. It reminds me of John Baez's "my favorite numbers" articles. $\endgroup$
    – Tim Campion
    Commented Nov 11 at 5:37

13 Answers 13

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The free modular lattice on $n$ generators is finite if $n\leq 3$ and infinite if $n\geq 4$.

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The word problem is decidable in every $\leq3$-manifold group, but can be undecidable in $4$-manifold groups.

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There are no genuinely quantum permutations on $N\leq 3$ symbols but there are from $N=4$.

The quantum permutation group $S_N^+$ on $N$ symbols is a virtual object given the unital $\mathrm{C}^*$-algebra $C(S_N^+)$ generated by the entries of an $N\times N$ magic unitary. A magic unitary is a matrix with entries in a unital *-algebra whose entries are orthogonal projections and whose rows and columns sum to the identity.

This algebra admits a compact quantum group structure.

For $N\leq 3$, the algebra is commutative and in fact by Gelfand theory isomorphic to the algebra of continuous (aka all) functions on the symmetric group. From $N=4$ the algebra is noncommutative and we have a "genuine" quantum group.

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    $\begingroup$ Please could you add a link or define quantum permutation? $\endgroup$
    – Mark Wildon
    Commented Nov 12 at 18:25
  • $\begingroup$ @MarkWildon added a bit there. $\endgroup$
    – JP McCarthy
    Commented Nov 12 at 19:00
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For $n\leq3$, all groups of order $n$ are isomorphic, but there are two nonisomorphic groups of order $4$.

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Let $K$ be an infinite field. Fix $N \ge 2$ and consider $n$-tuples of subspace of a vector space $K^N$ up to the diagonal action of $\mathrm{GL}_N(K)$. There are finitely many isomorphism classes if $n \le 3$ and infinitely many whenever $n \ge 4$.

In the language of representation theory, the star quiver with $n$ leaves has finite representation type if and only if $n \le 3$.

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    $\begingroup$ Is this related to the undirected graph of the star quiver being a Dynkin diagram up to $n = 3$, but not after? $\endgroup$
    – LSpice
    Commented Nov 11 at 23:13
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    $\begingroup$ Yes. Gabriel's Theorem is that the module category of a quiver algebra over an infinite field has finite type if and only if the underlying undirected graph of the quiver is of Dynkin type A, D or E. $\endgroup$
    – Mark Wildon
    Commented Nov 12 at 17:29
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    $\begingroup$ Maybe worth noting that for $N = 2$ this is just the fact that any $3$ points in $\mathbb{P}^1$ are projectively equivalent, but for $4$ points the cross-ratio is an invariant. $\endgroup$
    – Nate
    Commented Nov 12 at 20:40
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    $\begingroup$ Note that $n=4$ is also qualitatively different from $n\ge 5$ since the star quiver with four leaves is an affine Dynkin diagram, so has tame representation type (while the representation type is wild for $n\ge 5$). $\endgroup$
    – Antoine Labelle
    Commented Nov 15 at 5:02
  • $\begingroup$ Showing that, when the next iteration of this question comes around, we can copy and paste this answer with minor changes. $\endgroup$
    – Mark Wildon
    Commented Nov 15 at 17:38
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Riemannian geometry in dimensions 4 and higher is often quite different compared to two- and three-dimensional geometry, due to the Ricci decomposition. For surfaces, the entire Riemann curvature tensor can be determined from the scalar curvature alone. In three dimensions, the Ricci curvature suffices to determine the entire Riemann curvature tensor. However, in higher dimensions there is the Weyl tensor, which plays an important role in the analysis of higher-dimensional manifolds.

It's probably not possible to give a complete list for all the places where this distinction manifests itself, but one example is in the analysis of Ricci flow. In three-dimensions, the eigenvalues of the Ricci tensor evolve in a relatively well-controlled way, which gives rise to an effect known as Hamilton-Ivey pinching. This plays a crucial role in understanding the possible geometry of singularities, but fails in higher dimensions (unless one imposes much stronger assumptions on the geometry).

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In ordinal analysis, an analysis of the theory $\mathrm{KP}+\Pi_3\text{-reflection}$ was first published by Rathjen in 1994 (Rathjen, "Proof Theory of Reflection"), the function on ordinals used to obtain the ordinal notation has a definition (definition 4.8) that only is about a third of a page long. At the end of this paper, Rathjen provides a similarly simple ordinal function for a future analysis of $\mathrm{KP}+\Pi_4\text{-reflection}$, but as far as I know this turned out to be incorrect as the function used did not account for complexities that arose when trying to carry out the analysis.

Since then, all published definitions become much more complex when attempting to analyze $\mathrm{KP}+\Pi_4\text{-reflection}$. Some analyses exist, however the ordinal functions appearing in them have much longer definitions, for example:

  • In Duchhardt's 2008 dissertation Thinning Operators and $\Pi_4$-Reflection, the definition (pp. 17–19) is just over two pages.
  • In Arai's later paper "A simplified ordinal analysis of first-order reflection", the definition (definitions 2.4–2.5) is about one and a half pages, or two and a half if including the preliminary definitions of definition 2.1 which are not standard material. (This is a stronger function defined in aiming for a stronger result than ordinal analysis of $\mathrm{KP}+\Pi_4\text{-reflection}$, but the $N=4$ case will give the desired result and a definition of the same length.)

This complexity increase has been remarked on by a few different people:

An ordinal analysis of $\Pi_\omega\text-\mathsf{Ref}$ requires a transfinite iteration of techniques needed for an ordinal analysis of $\Pi_3\text-\mathsf{Ref}$ as published in [Rat94b]. The ordinal analysis of $\Pi_4\text-\mathsf{Ref}$ given by Christoph Dichhardt in [Duc08] reveals that even a finite iteration of these techniques is far from being trivial.

~ Stegert, in his 2010 dissertation Ordinal proof theory of Kripke-Platek set theory augmented by strong reflection principles. His dissertation gives a five-page definition of an ordinal function, this would have been the longest in this answer, however I decided to omit it as it is for analyzing the stronger system $\Pi_\omega\text-\mathsf{Ref}$ only, unlike Arai's which may be restricted to an $N=4$ case.

The downside of Rathjen's approach is … additional complexity when trying to extend it (especially beyond $\Pi_3$-reflection) …. Even more recent systems along these lines are fairly complex. For examples, see (Duchhardt 2008) for $\Pi_4$ reflection, (Arai 2015) for $\Pi_n$ reflection, and (Stegert 2010) for stability.

~ Taranovsky giving an overview of the (at the time) state of the art in ordinal analysis, on his webpage "Ordinal Notation".

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The Lie algebra $\operatorname{so}(4)$ splits into a direct product; $\operatorname{so}(3)$ doesn't.

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    $\begingroup$ Can you add what you mean by $\operatorname{so}(n)$? Because it is lowercase it might now be as universally known. $\endgroup$
    – M. Winter
    Commented Nov 15 at 0:03
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    $\begingroup$ @M.Winter, I think lowercase (or lowercase Fraktur/Sütterlin) is reasonably common for Lie algebras, which may have been why the poster left it out, but I went ahead and edited to remove any confusion. $\endgroup$
    – LSpice
    Commented Nov 15 at 15:11
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On page 12 of the IHES preprint version of "Chirurgie des Grassmanniennes" by Laurent Lafforgue, he mentions that a certain morphism related to compactifications of ${\rm PGL}_{n}^{4}/{\rm PGL}_n$ is smooth for $n\le 3$ and then fails to be flat for $n=4$ and above.

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When g ≤ 3, every principally polarized abelian variety of dimension g is the Jacobian of a curve of genus g, but most are not for g ≥ 4.

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For each $d \le 3$, there are plenty of smooth projective plane curves of degree $d$ with infinitely many rational points, but for $d \ge 4$ there are none.

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    $\begingroup$ Cute, although slightly cheating, since it's really a change from $$g=\frac{(d-1)(d-2)}{2}\le1$$ to $$g=\frac{(d-1)(d-2)}{2}\ge2.$$ So it's a hidden $2$ to $3$ change. $\endgroup$
    – Joe Silverman
    Commented Nov 15 at 15:34
  • $\begingroup$ @Joe, of course! Like my answer to the related 2 vs. 3 question asked recently, I just figured I'd post something simple and not to be taken too seriously. $\endgroup$
    – John Doyle
    Commented Nov 15 at 18:49
  • $\begingroup$ Typo (pointed out by John), I should have said a hidden "1 to 2" change. $\endgroup$
    – Joe Silverman
    Commented Nov 15 at 20:18
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Distinct tilings of space by both the measure polytope and the cross polytope exist only in four dimensions (the measure and cross polytopes are identical in one or two dimensions).

Measure polytopes always tile space, but cross polytopes do not. For instance, in three dimensions cubes tile space but regular octahedra do not (unless supplemented by regular tetrahedra). In $n$ dimensions, the dihedral angle $\theta$ between adjacent cells satsifies $\cos\theta=(2/n)-1$; this corresponds to $\theta$ dividing $360°$ evenly only for $n=1$ or $2$ (where as mentioned the cross polytope reduces to the measure polytope) and for $n=4$.

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4 is the smallest composite number.

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  • $\begingroup$ I do not know why this took so long. $\endgroup$
    – Mauricio
    Commented Dec 8 at 17:55

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