Toda–Smith complex

In mathematics, Toda–Smith complexes are spectra characterized by having a particularly simple homology, and are used in stable homotopy theory.

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.

The story begins with the degree ${\displaystyle p}$ map on ${\displaystyle S^{1}}$ (as a circle in the complex plane):

The degree ${\displaystyle p}$ map is well defined for ${\displaystyle S^{k}}$ in general, where ${\displaystyle k\in \mathbb {N} }$. If we apply the infinite suspension functor to this map, ${\displaystyle \Sigma ^{\infty }S^{1}\to \Sigma ^{\infty }S^{1}=:\mathbb {S} ^{1}\to \mathbb {S} ^{1}}$ and we take the cofiber of the resulting map:

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