I have a map, constructed geometrically, $S^4 \to S^3$. I suspect that it is a representative for the generator $\eta_3\in \pi_4(S^3) \simeq \mathbb{Z}_2$, but I am not 100% sure ($\eta_3$ is defined as the suspension of the Hopf map $S^3 \to S^2$). I'd like to be able to detect the fact that my map is homotopically nontrivial in a geometric or combinatorial way. Is this possible? I imagine there is some sort of index one can calculate, but unfortunately the terms one looks these things up in are very generic (forms, integration, index etc).
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How about thinking about framed cobordism, which in this case gives an isomorphism between $\pi_4(S^3)$ and the group of cobordism classes of normally framed 1manifolds in $S^4$. Since your map is constructed geometrically, it probably has a regular value. Pull this back to a collection of disjoint circles in $S^4$, forming a trivial link, with framings of their normal bundles. Since $\pi_1SO(3)$ has order 2, just count up the number of circles with nontrivial framing to see whether this number is odd or even. 

