nLab toda-smith complex

Informal definition

Toda-Smith complexes are finite 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 (localized at a prime) into families of maps via chromatic homotopy theory.

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 $k \in math{N}$. 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 $* \neq n$).

Formal definition

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

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:

• $BP_*(V(-1)) \simeq BP_*$
• $BP_*(V(0)) \simeq BP_*/p$
• $BP_*(V(1)) \simeq BP_*/(p, v_1)$
• etc.

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 $V(0)$

Relevance to stable homotopy theory

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

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.

(… add in their relation to the $v_2$ periodic elements, and the reasoning behind the shift $2(p^i-1)$)