The study of $n$-manifolds has some well-known periodicities in $n$ with period a power of $2$:
- $n \bmod 2$ is important. Poincaré duality implies that odd-dimensional compact oriented manifolds have Euler characteristic zero, while even-dimensional compact oriented manifolds have a middle cohomology group which inherits a nondegenerate pairing given by the cup product. The orthogonal groups $\text{O}(n)$ and the spheres $S^n$ behave differently when $n$ is even vs. when $n$ is odd. Only when $n$ is even do we have symplectic or complex manifolds, and Chern numbers only exist when $n$ is even.
- $n \bmod 4$ is important. In even dimensions $n = 2k$, the nondegenerate pairing on middle cohomology given by the cup product is symplectic if $k$ is odd but symmetric if $k$ is even; in the latter case this lets us define the signature, and in the former case, the additional data of a framing lets us define the Kervaire invariant. Pontryagin numbers only exist when $n$ is divisible by $4$.
- $n \bmod 8$ is important. The existence of the Atiyah-Bott-Shapiro orientation $\text{MSpin} \to KO$ implies that there are $8$-fold periodicity phenomena in the study of spin manifolds coming from Bott periodicity. The induced map on homotopy groups refines the $\widehat{A}$ genus, which it reproduces when $n$ is divisible by $4$, but we also get two extra $\mathbb{Z}_2$-valued invariants when $n$ is $1, 2 \bmod 8$.
- The binary expansion of $n$ is important. The number $\alpha(n)$ of $1$s in the binary expansion of $n$ controls how many Stiefel-Whitney classes of the stable normal bundle automatically vanish via Wu's formula and consequently figures in Cohen's immersion theorem that any smooth compact $n$-manifold immerses into $\mathbb{R}^{2n - \alpha(n)}$ ($n \ge 2$).
For some periods that are even but not powers of $2$, I have the impression that there ought to be phenomena of period $24$ or maybe even $576$ in the study of string manifolds coming from the $\sigma$-orientation $\text{MString} \to \text{tmf}$, although I don't know anything concrete about this. Presumably that $24$ has something to do with the Leech lattice. And in chromatic homotopy theory there is $v_k$-periodicity, which I again don't know anything concrete about; this has period $2p^k - 2$ where we are working $p$-locally.
Are there interesting periodicities in $n$ with odd period? If not, are there good reasons not to expect them?
For example, are there interesting manifold invariants which are naturally only defined when $n$ is divisible by $3$?
