Let $f: S^m \rightarrow S^n$ be nullhomotopic, where $m \lt n$.
I'm trying to get an intuitive understanding or of what this means for $(m,n) = (0,1),(0,2), \text{ or }(1,2)$.
$(m,n) = (0,1)$: Two beads on a circle can be pushed to each other and fused or vice versa.
$(m,n) = (0,2)$: Two beads on a sphere can be pushed to each other and fused or vice versa.
$(m,n) = (1,2)$: Any loop on a sphere can be shrunk to a point inside the loop.
Does this make sense or are these comparisons misunderstanding a concept?
In general, a nullhomotopy intuitively means you can homotope the image to a point (a constant map). One way to intuitively understand this is that $f : S^m \to S^n$ is nullhomotopic if there is only a trivial way of 'wrapping' $S^m$ on $S^n$ (although this analogy is better when $n \geq m $). You can 'shrink' two dots ($S^0$) to a point, and you can shrink a circle to a point on $S^k, k \geq 2$. Note this explains the intuition behind other homotopy groups of spheres too. For example, the Hopf map from $S^3 \to S^2$ basically says that there is a non-trivial way of wrapping the 3-sphere around the 2-sphere. Does this make sense?