Elements in the homotopy groups of spheres leave a genealogical record through the EHP sequences, both on the geometric level and on the algebraic level in the lambda algebra. We approach this with the periodic lambda algebra. It is both conceptually simpler and smaller than the classical lambda algebra and has unstable approximations with EHP sequences.
The Smith-Toda complexes V(m), when they exist, provide a simplification of stable homotopy by removing lower periodic classes. There are even smaller versions of the periodic lambda algebra for the Smith-Toda complexes. On the algebraic level, there are unstable approximations to these lambda algebras with EHP sequences providing a genealogical development for V(m) as well.
We also consider the question of geometric models for the approximations to V(0) and V(1), which leads to a simple lambda algebra for the 3 sphere.