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:
- Fei Han, Ruizhi Huang, Kefeng Liu, Weiping Zhang, Cubic forms, anomaly cancellation and modularity.
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.
