In mathematics, Toda–Smith complexes are spectra characterized by having a particularly simple BP-homology, and are useful objects in stable homotopy theory.
Toda–Smith complexes provide examples of periodic self maps.
These self maps were originally exploited in order to construct infinite families of elements in the homotopy groups of spheres.
Their existence pointed the way towards the nilpotence and periodicity theorems.
[1] The story begins with the degree
p
map on
(as a circle in the complex plane): The degree
map is well defined for
in general, where
If we apply the infinite suspension functor to this map,
and we take the cofiber of the resulting map: We find that
has the remarkable property of coming from a Moore space (i.e., a designer (co)homology space:
is trivial for all
It is also of note that the periodic maps,
α
β
γ
, come from degree maps between the Toda–Smith complexes,
th Toda–Smith complex,
, is a finite spectrum which satisfies the property that its BP-homology,
, is isomorphic to
That is, Toda–Smith complexes are completely characterized by their
{\displaystyle BP}
-local properties, and are defined as any object
satisfying one of the following equations: It may help the reader to recall that
deg