Maps with Hopf invariant zero are suspensions

Question

Let $h:\pi_{2n-1}(S^n) \rightarrow \mathbb{Z}$ be the Hopf invariant. I believe that in the same paper that proves his suspension theorem, Freudenthal proved that if $x \in \pi_{2n-1}(S^n)$ satisfies $h(x)=0$, then $x$ is in the image of the suspension map $\pi_{2n-2}(S^{n-1}) \rightarrow \pi_{2n-1}(S^n)$. Observe that the usual Freudenthal suspension theorem says that the map $\pi_{2n-1}(S^n) \rightarrow \pi_{2n}(S^{n+1})$ is a surjection.

Can anyone either describe a proof of this or point me in the direction of a modern source for it? The only one I know of is Pontryagin's book "Smooth manifolds and their applications in homotopy theory", where this is Theorem 16, but I find his proof very hard to follow. I guess I would particularly appreciate a proof in the spirit of Pontryagin's proof, though more algebraic proofs are welcome too.

Answer 1

In my notes "Homotopy groups of spheres and low-dimensional topology" (available on my page of notes here), I have written up a modern account of Pontryagin's approach to calculating the homotopy groups of spheres. In particular, Section 9 contains a detailed account of Pontryagin's proof of the theorem of Freudenthal you ask about.

Answer 2

The integral statement is most generally this:

For a based CW space $X$ one has a suspension map $E: X\to \Omega \Sigma X$ and a Hopf invariant $H: \Omega \Sigma X \to \Omega \Sigma (X\wedge X)$ (construction outlined below).

The composite $H\circ E$ is canonically null and the sequence $$ X \overset E \to \Omega \Sigma X \overset H \to \Omega \Sigma (X \wedge X) $$ is a homotopy fiber sequence in a range, roughly $3r$, where $r$ is the connectivity of $X$. Your result will follow from this easily (take $X = S^n$).

The map $H$ is often constructed as follows: let $JX$ be the reduced free monoid on the points of $X$. Using say the Moore loops model for $\Omega \Sigma X$, one can construct a monoid homomorphism $JX \to \Omega \Sigma X$ that extends the map $E$ using the universal property of the free monoid. A homology calculation shows that this map his a weak equivalence (James did this calculation when $X$ is a sphere).

Finally, the proof the above sequence is a fibration in the range roughly thrice the connectivity of $X$ can be deduced from the following four facts:

1) $J_2 X \to JX$ is $(3r+2)$-connected, where $J_2X = X \cup (X \times X)$ is filtration two.

2) The quotient $J_2X/X$ is $X\wedge X$, so we have a cofiber sequence $$ X \to J_2 X \overset q\to X \wedge X $$

3) By the Blakers-Massey Theorem, the above sequence is a homotopy fibration in the range roughly $3r$.

4) The diagram $$ \require{AMScd} \begin{CD} J_2X@>q>>X\wedge X\\ @VVV @VVV\\ JX@>>H>\Omega\Sigma(X\wedge X) \end{CD} $$ is commutative, where the left vertical maps is the inclusion and the right one is the suspension map for $X\wedge X$.

Answer 3

There is a $2$-local fiber sequence

$$S^n \to \Omega S^{n+1} \to \Omega S^{2n+1}$$

where the first map is the suspension map. Its associated long exact sequence of homotopy groups is called the EHP sequence and the induced map $\pi_{2n} \Omega S^{n+1} \to \pi_{2n} \Omega S^{2n+1}$ coincides with the Hopf invariant $\pi_{2n+1} S^{n+1} \to \mathbb{Z}$. This answers your question and yields a lot more information about homotopy groups of spheres.

Good modern references are Hatcher's Spectral Sequences in Algebraic Topology and Neisendorfer's Algebraic Methods in Unstable Homotopy Thoery. There are also versions of the EHP sequence at odd primes but they are more subtle, they are discussed in Ravenel's Complex Cobordism and Stable Homotopy Groups of Spheres.