# What results about the topology of manifolds depend on the dimension mod 3?

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.

• 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
Commented Jan 9, 2022 at 6:12
• Commented Jan 9, 2022 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. Commented Jan 9, 2022 at 13:01
• "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. Commented Jan 10, 2022 at 11:38
• 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. Commented Jan 11, 2022 at 18:59

I learned recently that the 'topological modular form' cohomology theory TMF has $$576$$-fold periodicity. Not quite period $$3$$, but similar, since $$576=3^2 \times 2^6$$.
• The world would be a really boring place without the primes $2$ and $3$... Commented Jan 12, 2022 at 1:37
• For those who want to learn what $\pi_*(TMF)$ is without learning what $TMF$ is, I can recommend my survey chapter christopherleedouglas.com/wp-content/uploads/2015/10/…, starting at page 211. More precisely: pages 211-214. Commented Jan 12, 2022 at 14:49
• 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. Commented Jan 12, 2022 at 15:04