nLab
periodic map

A periodic map is a self-map of a spectrum XX of the form

Σ dXfX\Sigma^d X \xrightarrow{f} X

for some d0d \geq 0, the condition of periodicity is that, when we iterate ff, no Σ tdX\Sigma^{t d}X is null homotopic. (note that it’s increasing dimension by d):

...Σ 2dXΣ dfΣ dXfX ... \to \Sigma^{2d} X \xrightarrow{\Sigma^d f} \Sigma^d X \xrightarrow{f} X

A related concept: If t\exists t s.t. Σ tdX\Sigma^{t d} X is null homotopic, then ff is instead called a nilpotent map.

A verbose introduction to periodic maps

Let us take the mindset of a harmonic analyst as we are handed one period of an interesting function S dXvXS^d X \xrightarrow{v} X. Our first inclination is to shift this map a step dd to the left and lay them next to each other, ad infinitum.

...S 3dvS 3dXS 2dvS 2dXS dvS dXvX... \xrightarrow{S^{3d} v} S^{3d} X \xrightarrow{S^{2d}v} S^{2d} X \xrightarrow{S^d v} S^d X \xrightarrow{v} X

for notational simplicity:

...v 4S 3dXv 3S 2dXv 2S dXvX... \xrightarrow{v^4} S^{3d} X \xrightarrow{v^3} S^{2d} X \xrightarrow{v^2} S^d X \xrightarrow{v} X

If, when we iterate vv enough times, we find ourselves looking at a sad contractible space, tired by our repetitive antics, shriveled to a point, we quite naturally call vv nilpotent. But we are harmonic analysts – seekers of periodicity! We wish to look at vv which do not die when we suspend them over and over, those that are periodic with period dd. So…in what case does vv not die?

With vv on our mind we look at an element ff in the homotopy classes of maps from S kXS^k X to YY, represented by the map:

S kXfYS^k X\xrightarrow{f} Y

How do we fit this together with our self map vv?

S dXvXS^d X \xrightarrow{v} X

We shift vv

S d(S kX)S kv(S kX)S^d(S^k X) \xrightarrow{S^k v} (S^k X)

and lay it in place next to ff.

...S kv 3S 2d(S kX)S kv 2S d(S kX)S kv(S kX)fY... \xrightarrow{S^k v^3} S^{2d}(S^k X) \xrightarrow{S^k v^2} S^d(S^k X) \xrightarrow{S^k v}(S^k X) \xrightarrow{f} Y

We might look at this collection of maps,

S jd(S kX)fS kv jYS^{j d}(S^k X) \xrightarrow{f \circ S^k v^j} Y

In slightly more palatable notation, we’re looking at the collection v jf[S jd+kX,Y]v^j f \in [S^{j d+k} X, Y] as jj ranges over the natural numbers – forbidding ourselves from looking at vv if v jv^j is constant for any jj.

Why limit ourselves to S jd+kS^{j d+k}? Let us look at all S *:=S n nS^* := {S^n}_{n \in \mathbb{Z}}, that is, we look at the set v jf[S *X,Y]v^j f \in [S^* X, Y] (still demanding that all v jfv^j f are nontrivial):

{f,vf,v 2f,...}\{f, v f, v^2f, ...\}

We name this set of vv-periodic elements in [S *X,Y][S^*X, Y] a “vv-periodic family”

Note: The v nv_n-periodic families are not actually periodic in π * S\pi_*^S, they are maps induced by periodic maps between CW-complexes.

Examples of periodic maps

See toda-smith complex.

An example motivating the study of periodic maps: the bott element in kuku. (I’m not sure that this is entirely correct).

S 2βkuS^2 \xrightarrow{\beta} ku
S 2kukukukuS^2 \wedge ku \to ku \wedge ku \to ku
Σ 2ku×βku\Sigma^2 ku \xrightarrow{\times \beta} ku

Last revised on June 10, 2016 at 14:06:28. See the history of this page for a list of all contributions to it.