Tools of Unstable Homotopy Theory

3 The EHP Sequence

The EHP sequence is a fibre sequence due to James at the prime \(2\) and Toda at odd primes, which relates the homotopy groups of different spheres. One can also view it as giving an understanding of the spectral sequence of the tower \(\Omega ^n S^n\) approximating \(QS^0\).

At the prime \(2\), we will identify the fibre of the second James-Hopf map \(H_2: \Omega \Sigma S^n \to \Omega \Sigma S^{2n}\) with \(S^n\).

Theorem 3.1. There is a \(2\)-local fibre sequence \(S^n \xrightarrow {E}\Omega \Sigma S^n \xrightarrow {H} \Omega \Sigma S^{2n}\) where \(H\) is the James-Hopf map \(H_2\). For \(n\) odd, one need not \(2\)-localize.

Proof. First, we claim that \(H\) is an isomorphism on \(H_{2n}\). To see this, \(\Sigma H\) factors as \(\Sigma \Omega \Sigma S^n \to \Sigma S^{2n} \to \Sigma \Omega \Sigma S^{2n}\), where the first map is the projection from the James splitting, and the second is the counit. Both of these maps are isomorphisms on \(H_{2n+1}\).

Next, we claim that atleast \(2\)-locally, \(H_2\) is an isomorphism in cohomology in all degrees where it is nonzero. For \(n=2k\) even, we know that the generator \(y_{4k}\) of \(H^*(\Omega \Sigma S^{4k})\) is sent to \(x_{2k}^{(2)}\), the second divided power of the class in degree \(4n\) of \(H^*(\Omega \Sigma S^{2k})\). It follows that \(y_{4k}^{(l)}\) is sent to \(x_{2k}^{(2l)} \frac {(2l)!}{2^l l!}\), which is a unit multiple of the generaor \(x_{2k}^{(2l)}\) in that degree. It follows from the Serre spectral sequence that inclusion of the fibre on homology agrees with \(E\), but then the map must just be \(E\).

For \(n=2k+1\), the divided power generator gets sent to a divided power generator, and so again one sees via the Serre spectral sequence that the fibre is \(S^n\), but this time integrally. □

The \(P\) in the EHP sequence is for Whitehead product, and refers to the associated map \(P:\Omega ^2\Sigma S^{2n} \to S^n\) on the bottom cell is the Whitehead square of the identity, which is easy to see from the definition of the Whitehead product.

The fibre sequence \(S^n \xrightarrow {E}\Omega \Sigma S^n \xrightarrow {H} \Omega \Sigma S^{2n}\) splits at odd primes for \(n\) odd: the map \(\Omega [i_{n+1},i_{n+1}]\) gives a splitting up to isomorphism. One can multiply the section and the inclusion of the fibre to obtain:

Corollary 3.2 (Serre). After inverting \(2\), there is an equivalence \(S^{2k-1}\times \Omega S^{4k-1} \simeq \Omega S^{2k}\).

The EHP sequence for \(n\) odd doesn’t split at the prime \(2\) unless there is an element of Hopf invariant \(1\).

Let \(J_{i}X\) be the \(i\)th term in the James filtration on the free monoid on \(X\). The odd prime extension of the EHP sequence is below.

Theorem 3.3. There are \(p\)-local fibre sequences:

\[ J_{p-1}S^{2n} \to \Omega \Sigma S^{2n} \xrightarrow {H_p} \Omega \Sigma S^{2np}\]

\[ S^{2n-1} \to \Omega J_{p-1} S^{2n} \to \Omega \Sigma S^{2np-2}\]