Toda–Smith complexes provided examples of periodic maps. Thus, they led to the construction of the nilpotent and periodicity theorems, which provided the first organization of the stable homotopy groups of spheres into families of maps localized at a prime.
Mathematical context
The story begins with the degree p map on S1 (as a circle in the complex plane):
S1→S1z↦zp
The degree p map is well defined for Sk in general, where k∈N. If we apply the infinite suspension functor to this map, Σ∞S1→Σ∞S1=:S1→S1 and we take the cofiber of the resulting map:
S→pS→S/p
We find that S/p has the remarkable property of coming from a Moore space (i.e., a designer (co)homology space: Hn(X)≃Z/p, and H~∗(X) is trivial for all ∗≠n).
It is also of note that the periodic maps, αt, βt, and γt, come from degree maps between the Toda–Smith complexes, V(0)k, V(1)k, and V2(k) respectively.
Formal definition
The nth Toda–Smith complex, V(n) where n∈−1,0,1,2,3,…, is a finite spectrum which satisfies the property that its BP-homology, BP∗(V(n)):=[S0,BP∧V(n)], is isomorphic to BP∗/(p,…,vn).
That is, Toda–Smith complexes are completely characterized by their BP-local properties, and are defined as any object V(n) satisfying one of the following equations:
