Toda–Smith complex

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

${\displaystyle S^{1}\to S^{1}\,}$

${\displaystyle z\mapsto z^{p}\,}$

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:

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

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

It is also of note that the periodic maps, ${\displaystyle \alpha _{t}}$, ${\displaystyle \beta _{t}}$, and ${\displaystyle \gamma _{t}}$, come from degree maps between the Toda–Smith complexes, ${\displaystyle V(0)_{k}}$, ${\displaystyle V(1)_{k}}$, and ${\displaystyle V_{2}(k)}$ respectively.

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

That is, Toda–Smith complexes are completely characterized by their ${\displaystyle BP}$-local properties, and are defined as any object ${\displaystyle V(n)}$ satisfying one of the following equations:

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

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

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