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{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 $BP_{*}={\mathbb {Z}}_{p}[v_{1},v_{2},...]$ , $\deg v_{i}$ = $2(p^{i}-1)$ .

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

↑ 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

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