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).



Detecting homotopy nontriviality of an element in a torsion homotopy group
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 1-manifolds 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. 


