# Are there periodicity phenomena in manifold topology with odd period?

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 $3$?

• A better question might be: what are some periodicity phenomena in manifold topology with period not a power of two? – André Henriques Oct 31 '14 at 15:43
• There are lots of maps involving the surgery exact sequence which are isomorphisms away from the prime $2$ but which do strange things at $2$. A big part of this is of course that quadratic forms over group algebras behave strangely in characteristic $2$. I wonder if this is the underlying explanation for your observations about periodicity. – Paul Siegel Nov 1 '14 at 23:22
• @Urs: well, in the context of periodic ring spectra there's a straightforward reason to favor even periodicity: if any element $\beta$ of odd degree in a graded commutative ring is invertible with respect to multiplication, then $\beta^2$ being $2$-torsion implies the entire ring is $2$-torsion. I guess more geometrically this argument suggests the evenness is about the graded commutativity of the intersection pairing. – Qiaochu Yuan Nov 2 '14 at 9:02
• A 12 mod 16 comes up in a cobordism group. See the comments here: mathoverflow.net/questions/165609/… – Ryan Budney Nov 6 '14 at 14:37
• Are these periodicities reflected in, or reflections of, non-nilpotent elements in some graded commutative rings? If so, the period would have to be even. – John Palmieri Nov 12 '14 at 18:03