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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 $$ J_2X \,\, \overset q\to X\wedge X $$ $$ \downarrow \qquad \quad\,\,\downarrow $$ $$ JX \underset H\to \Omega \Sigma (X\wedge X) $$ is commutative, where the left vertical maps is the inclusion and the right one is the suspension map for $X\wedge X$.

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 $X\wedge X$.

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 $3r$-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 $$ J_2X \,\, \overset q\to X\wedge X $$ $$ \downarrow \qquad \quad\,\,\downarrow $$ $$ JX \underset H\to \Omega \Sigma (X\wedge X) $$ is commutative, where the left vertical maps is the inclusion and the right one is the suspension map for $X\wedge X$.

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 $$ J_2X \,\, \overset q\to X\wedge X $$ $$ \downarrow \qquad \quad\,\,\downarrow $$ $$ JX \underset H\to \Omega \Sigma (X\wedge X) $$ is commutative, where the left vertical maps is the inclusion and the right one is the suspension map for $X\wedge X$.

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 $JX \to \Omega \Sigma X$ extends the map $X$ 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 $3r$-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 $$ J_2X \,\, \overset q\to X\wedge X $$ $$ \downarrow \qquad \quad\,\,\downarrow $$ $$ JX \underset H\to \Omega \Sigma (X\wedge X) $$ is commutative, where the left vertical maps is the inclusion and the right one is the suspension map for $X\wedge X$.

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 $3r$-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 $$ J_2X \,\, \overset q\to X\wedge X $$ $$ \downarrow \qquad \quad\,\,\downarrow $$ $$ JX \underset H\to \Omega \Sigma (X\wedge X) $$ is commutative, where the left vertical maps is the inclusion and the right one is the suspension map for $X\wedge X$.