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 nth 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

References

This article is issued from Wikipedia - version of the Thursday, December 17, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.