Hopf invariant

Remarks

The integer $H(f)$ depends only on the homotopy class of $f$, and $H$ is a homomorphism $\pi_{2n-1}(S^n) \to \Z$ on homotopy groups. A change of the generator $\beta$ changes only the sign of $H(f)$.

Example: for the Hopf map $h \colon S^3 \to S^2$, the space $C_h$ is $\C P^2$. The square of a generator of $H^2(\C P^2; \Z)$ generates $H^4(\C P^2; \Z)$, so $H(h) = \pm 1$.

Non-example: for $n = 1$ the construction breaks down. A map $S^1 \to S^1$ of degree $d$ gives $H^2(C_f; \Z) \cong \Z/d$, which is not $\Z$ when $d \ne 0$, so no generator $\beta$ as above exists.

If $n$ is odd, then $\alpha \smile \alpha = -\alpha \smile \alpha$ by graded commutativity, and since $H^{2n}(C_f; \Z) \cong \Z$ has no torsion, $H(f) = 0$. If $n$ is even, the Whitehead product $[\iota_n, \iota_n]$ of the identity of $S^n$ with itself has Hopf invariant $\pm 2$.