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There are a lot of interesting results about the topology of manifolds that depend on the dimension of the manifold mod 2, mod 4, or mod 8. The simplest ones involve the cup product

$$ \smile \colon H^{n/2}(M,R) \times H^{n/2}(M,R) \to H^{n}(M,R) \cong R$$

where $n$, the dimension of $M$, is even, $R$ is a commutative ring, and the last isomorphism is present if $M$ is compact and oriented (or more generally $R$-oriented).

When $n$ is a multiple of $4$, this trick gives a nondegenerate symmetric bilinear form

$$ \smile \colon H^{n/2}(M,\mathbb{R}) \times H^{n/2}(M,\mathbb{R}) \to \mathbb{R}$$

and the signature of this bilinear form is an important invariant called the signature of $M$. In this case we also get a lattice $L$ that's the image of $H^{n/2}(M,\mathbb{Z})$ in $H^{n/2}(M,\mathbb{R})$, and a unimodular symmetric bilinear form

$$ \smile \colon L \times L \to \mathbb{Z}$$

which gives more refined information about $M$. This is very important in the classification of compact oriented 4-dimensional manifolds.

When the dimension of $M$ equals 2 mod 4, we instead get a nondegenerate skew-symmetric bilinear form

$$ \smile \colon H^{n/2}(M,\mathbb{R}) \times H^{n/2}(M,\mathbb{R}) \to \mathbb{R}$$

otherwise known as a symplectic structure, and a unimodular skew-symmetric bilinear form

$$ \smile \colon L \times L \to \mathbb{Z}.$$

These data are not directly helpful in classifying manifolds because they're determined up to isomorphism by the dimension of $H^{n/2}(M,\mathbb{R})$, unlike in the case where $n$ is a multiple of $4$. But if $M$ is equipped a 'framing', then we can get an interesting invariant by taking $R = \mathbb{Z}/2$ and improving the bilinear form above to a quadratic form. This is called the Kervaire invariant.

There are subtler tricks involving spinors that depend heavily on the dimension of the manifold mod 8. There's also a nice result that depends on the dimension of the manifold mod $2^n$ for all $n$: namely, any smooth compact $n$-dimensional manifold admits an immersion into Euclidean space of dimension $2n-H(n)$, where $H(n)$ is the number of 1's in the binary expansion of $n$. (See Cohen's paper The immersion conjecture for differentiable manifolds.)

All this made me wonder if there are interesting results about manifolds that depend on their dimension mod 3 — or for that matter, any number that's not just a power of 2. Do you know any?

If $M$ is oriented and its dimension is a multiple of 3, we can use the cup product to get a trilinear form

$$ \smile \colon H^{n/3}(M,\mathbb{R}) \times H^{n/3}(M,\mathbb{R}) \times H^{n/3}(M,\mathbb{R}) \to \mathbb{R}$$

and if $L$ is the image of $H^{n/3}(M,\mathbb{Z})$ in $H^{n/3}(M,\mathbb{R})$ this restricts to give a trilinear form

$$ \smile \colon H^{n/3}(M,\mathbb{Z}) \times H^{n/3}(M,\mathbb{Z}) \times H^{n/3}(M,\mathbb{Z}) \to \mathbb{Z}.$$

These are symmetric when $n$ is a multiple of 6 but skew-symmetric when $n$ equals 3 mod 6.

Of course this already is a partial answer to my question, and we could easily generalize to numbers other than 3. But do these trilinear forms give nontrivial invariants of $M$? Are they used for anything interesting?

In a quick attempt to search for this I bumped into a paper on cubic forms that are invariants of 12-dimensional manifolds:

Also, Ahmet Beyaz has a paper A new construction of 6-manifolds that builds on a paper (Classification problems in differential topology. V. On certain 6-manifolds) where C.T.C. Wall classified simply-connected, compact oriented 6-manifolds with spin structure and torsion-free cohomology with the help of the trilinear form

$$ \smile \colon H^{2}(M,\mathbb{Z}) \times H^{2}(M,\mathbb{Z}) \times H^{2}(M,\mathbb{Z}) \to \mathbb{Z}.$$

But I don't know if either of these is part of a bigger story that involves the manifold's dimension mod 3, or 6, or 12.

John Baez
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    For the special case of dimension 6, you might have a look at Cubic forms and complex $3$-folds by Okonek and Van de Ven, Enseign. Math. (2) 41 (1995), no. 3-4, 297–333. As the title indicates they are particularly interested in complex threefolds, but they have some more general remarks. – abx Jan 09 '22 at 06:12
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    A related question. – Michael Albanese Jan 09 '22 at 12:45
  • Your example with a triple product on $H^{n/3}$ is an instance of essentially mod $2$ phenomenon - it just says the "product" on $H^a\times H^a\times H^a$ is symmetric if all $a$ are even and antisymmetric otherwise. – Wojowu Jan 09 '22 at 13:01
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    "But do these trilinear forms give nontrivial invariants of M?" Yes. The first example is the discriminant of a symmetric cubic form, which is an integer. So there is an integer topological invariant $M \mapsto \Delta(M)$ which only makes sense for manifolds of dimension 6k. In dimension 6, when the manifold admits a complex structure there are some connections of the discriminant to its complex geometry, and a nice formula for the discriminant of a $\mathbb{CP}^1$-bundle over $\mathbb{CP}^2$, and more. See the paper mentioned above. – Nick L Jan 10 '22 at 11:38
  • "dimension mod 3, or 6, or 12." -- why not "dimension mod 3, or 9, or 27." ? – Wlod AA Jan 10 '22 at 12:11
  • Indeed, go ahead and pick your own favorite integers. I gave some very simple examples of how the existence of certain invariants depends on the dimension mod 3 or 6. – John Baez Jan 11 '22 at 00:23
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    My PhD student, Yang Hu, is about to post a paper about topological complex vector bundles over complex projective spaces (using Weiss calculus), and the answer depends on the dimension of the projective space modulo 24. – Dev Sinha Jan 11 '22 at 18:59

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I learned recently that the 'topological modular form' cohomology theory TMF has $576$-fold periodicity. Not quite period $3$, but similar, since $576=3^2 \times 2^6$.

user1504
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    Do I understand right that one can think of this as a very fancy version of the fact that elliptic curves are kind of annoying in characteristics 2 and 3? – Noah Snyder Jan 11 '22 at 20:39
  • Safe to say that both phenomena spring from the same source. – user1504 Jan 11 '22 at 20:45
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    @NoahSnyder Is "kind of annoying" a phrase that's synonymous with "fascinating and subtle?" :) – Joe Silverman Jan 11 '22 at 23:18
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    The world would be a really boring place without the primes $2$ and $3$... – Andy Putman Jan 12 '22 at 01:37
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    For those who want to learn what $\pi_*(TMF)$ is without learning what $TMF$ is, I can recommend my survey chapter http://www.christopherleedouglas.com/wp-content/uploads/2015/10/TMF-preliminary.pdf, starting at page 211. More precisely: pages 211-214. – André Henriques Jan 12 '22 at 14:49
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    The linked page contains the sentence "$\mathit{tmf}_n$ is the space of supersymmetric 2d conformal field theories of central charge -n". This is not the most up-to-date version of the conjecture. The more up-to-date version reads: "$\mathit{TMF}_n$ is the space of $\mathcal N$=(0,1) supersymmetric 2d quantum field theories of gravitational anomaly -n". The significant changes are tmf --> TMF, and CFT --> QFT. – André Henriques Jan 12 '22 at 15:04
  • Does the $24^2$-periodicity of TMF imply that the topology of $n$-manifolds is similar in some way to the topology to the topology of $(n+24^2)$-manifolds? If so, some details would be great. – John Baez Jan 12 '22 at 23:19
  • @JohnBaez I don't see a direct statement of the form 'w(X) only depends on $d(X) \mod 576$. So this might not be an answer. $TMF(X)$ is a $d$-dimensional family of QFTs with target space of dimension $n$, so there's still a step needed to link $d$ and $n$. I was hoping to spot something in the Witten genus, but no luck yet. – user1504 Jan 13 '22 at 00:07