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,^[1] 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 $S^1$ (as a circle in the complex plane):

$S^1 \to S^1 \,$

$z \mapsto z^p \,$

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

$S \xrightarrow{p} S \to S/p$

We find that $S/p$ has the remarkable property of coming from a Moore space (i.e., a designer (co)homology space: $H^n(X) \simeq Z/p$, and $\tilde{H}^*(X)$ is trivial for all $* \neq n$).

It is also of note that the periodic maps, $\alpha_t$, $\beta_t$, and $\gamma_t$, come from degree maps between the Toda–Smith complexes, $V(0)_k$, $V(1)_k$, and $V_2(k)$ respectively.

Formal definition

The $n$th Toda–Smith complex, $V(n)$ where $n \in -1, 0, 1, 2, 3, \ldots$, is a finite spectrum which satisfies the property that its BP-homology, $BP_*(V(n)) := [\mathbb{S}^0, BP \wedge V(n)]$, is isomorphic to $BP_*/(p, \ldots, v_n)$.

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:

$\begin{align} BP_*(V(-1)) & \simeq BP_* \\[6pt] BP_*(V(0)) & \simeq BP_*/p \\[6pt] BP_*(V(1)) & \simeq BP_*/(p, v_1) \\[2pt] & {}\,\,\,\vdots \end{align}$

It may help the reader to recall that that $BP_* = \mathbb{Z}_p[v_1, v_2, ...]$, $\deg v_i$ = $2(p^i-1)$.

Examples of Toda–Smith complexes

• the sphere spectrum, $BP_*(S^0) \simeq BP_*$, which is $V(-1)$.
• the mod p Moore spectrum, $BP_*(S/p) \simeq BP_*/p$, which is $V(0)$

References

1. https://books.google.com/books?id=xoM5DxQZihQC&pg=PA341&lpg=PA341&dq=Larry+Smith+algebraic+topology&source=bl&ots=pyH8hPMnye&sig=dVMAcZzrlqoTuKj010LhMe8dRIA&hl=en&sa=X&ved=0CDwQ6AEwBGoVChMI7af67tv3xgIVBzKICh0sbQWR#v=onepage&q=Larry%20Smith%20algebraic%20topology&f=false