Multiplicative structures on spheres

Ishan Levy

February 2, 2023

What multiplicative structures do the spheres \(S^n\) have? Globally, \(S^0 = \FF _2,S^1 = \text {B}\ZZ \) are \(\EE _\infty \)-spaces and \(S^3=\Omega \HH \PP ^\infty \) is a \(\EE _1\)-space. \(S^7\) admits an \(\AA _2\)-structure because of the octonions, but this doesn’t extend to \(\AA _3\), as there is a \(3\)-primary obstruction visible from the Steenrod algebra action on the cohomology of the hypothetical \(\mathbb {O}\PP ^3\). By Hopf invariant one, there are no more \(\AA _2\) multiplications on spheres, with the obstruction being \(K(1)\)-local at the prime \(2\).

If we work \(p\)-adically (equivalently \(p\)-locally), then more spheres can become multiplicative. Rationally, \(S^{2n}\) is not even \(\AA _2\) (the obstruction is the whitehead square of the identity), and since the obstruction persists after base-change to \(\QQ _p\), \(S^{2n}\) has no hope of being \(\AA _2\) \(p\)-adically. \(S^{2n-1}\) is an Eilenberg Mac Lane space rationally, so there is a hope of getting multiplicative structures \(p\)-adically for odd spheres.

Proposition 0.1. For \(p>2\), the map \(S^{2k-1} \to \Omega J_{p-1}(S^{2k})\) can be refined to a map of \(\AA _{p-1}\)-algebras.

Proof. The obstruction to refining an \(\AA _{i-1}\)-algebra map to an \(\AA _i\)-algebra map amounts to producing a lift in the diagram

where \(\text {ob}_i\) is the obstruction to refining an \(\AA _{i-1}\) algebra structure on \(S^{2k-1}\) to an \(\AA _i\) algebra structure.

By the EHP sequence, the fibre of the lower horizontal map is \(\Omega ^2 S^{2pk-1}\), so we learn that there is no obstruction to producing the lift for \(i<p\). □

From the proof above, we see that there is a potential obstruction to an \(\AA _p\)-structure on \(S^{2k-1}\). The obstruction doesn’t always cause issues:

Theorem 0.2 (Sullivan). \(S^{2k-1}\) is an \(\EE _1\)-algebra if \(k |p-1\).

Proof. For the first statement, we observe that \(S^{2k-1} = \Omega ((B^2\ZZ _p)_{hG_k})\), where \(G_k\) is the cyclic subgroup of \(\FF _p^\times \) of order \(k\) acting on \(\ZZ _p\). This can be seen from the Eilenberg-Moore/Serre spectral sequence. □

Wilkerson showed that in all other odd primary cases, there is a \(K(1)\)-local obstruction to an \(\AA _p\)-multiplication.

Theorem 0.3 (Wilkerson). If \(k \nmid p-1\), then \(S^{2k-1}\) is not \(\AA _p\).

Proof. If \(S^{2k-1}\) was \(\AA _p\), we could form the \(p^{th}\)-truncated bar construction \(B_p(S^{2k-1})\), which would have cohomology ring \(\ZZ _p[x]/x^{p+1}\). It follows that \(K^0(B_p(S^{2k-1}))\) as a ring is isomorphic to \(\ZZ _p[x]/x^{p+1}\). The Adams operations \(\psi ^q\), give an action of \(\QQ _p^{\times }\) on this, with the following properties:
- 1. \(\psi ^p(x) \equiv x^p \pmod {p}\)
- 2. \(\psi ^q(x) \equiv qx \pmod {x^2}\)
The second property comes from the inclusion of the first cell, which is a sphere, whose Adams operations we know.

Given a power series \(f\), let \(f[x^i]\) denote the \(x^i\)-coefficient. We show inductively for \(i \leq p\) that

\[v_p(\psi ^p(x)[x^i]) \geq k - (v_p(k)+1)|\{0<j<i \text { such that } p-1|jk\}|\]

Let \(m_i\) denote the number on the right hand side of the above equation. If we can show the claim, we are finished, since we know from \((1)\) that the \(x^p\) coefficient is \(1\) mod \(p\), but \(m_p = k- (v_p(k)+1)\gcd (i,p-1)\) is positive unless \(k|p-1\).

To show the claim inductively for \(i+1\) assuming it for lower \(i\), we compare the \(x^{i+1}\) coefficients of the equation \(\psi ^p\psi ^q(x)=\psi ^q\psi ^p(x)\) after reducing mod \(p^{m_i}\), using the inductive hypothesis to see that \(\psi ^p(x)[x^{i+1}]q^{k(i+1)} = \psi ^p(x)[x^{i+1}]q^{k}\) mod \(p^{m_i}\). It follows that \(v_p(\psi ^p(x)[x^i]) + v_p(q^{k(i+1)}-q^k) \geq m_i\). Choose \(q\) to be a topological generator of \(\ZZ _p^{\times }\), so that \(v_p(q^{k(i+1)}-q^n) \) is \(0\) unless \(p-1|k(i+1)\) in which case it is \(v_p(k)+1\). This gives the inductive step. □