Homotopy classes that are trivial mod $\mathcal{F}$

Abstract

If $\mathcal{ℱ}$ is a collection of topological spaces, then a homotopy class $\alpha$ in $\left[X,Y\right]$ is called $\mathcal{ℱ}$–trivial if

$${\alpha }_{\ast }=0:\left[A,X\right]\to \left[A,Y\right]$$

for all $A\in \mathcal{ℱ}$. In this paper we study the collection $Z_{\mathcal{ℱ}}\left(X,Y\right)$ of all $\mathcal{ℱ}$–trivial homotopy classes in $\left[X,Y\right]$ when $\mathcal{ℱ}=\mathcal{S}$, the collection of spheres, $\mathcal{ℱ}=\mathcal{ℳ}$, the collection of Moore spaces, and $F=\Sigma$, the collection of suspensions. Clearly

$$Z_{\Sigma }\left(X,Y\right)\subseteq Z_{\mathcal{ℳ}}\left(X,Y\right)\subseteq Z_{\S }\left(X,Y\right),$$

and we find examples of finite complexes $X$ and $Y$ for which these inclusions are strict. We are also interested in $Z_{\mathcal{ℱ}}\left(X\right)=Z_{\mathcal{ℱ}}\left(X,X\right)$, which under composition has the structure of a semigroup with zero. We show that if $X$ is a finite dimensional complex and $\mathcal{ℱ}=\mathcal{S}$, $\mathcal{ℳ}$ or $\Sigma$, then the semigroup $Z_{\mathcal{ℱ}}\left(X\right)$ is nilpotent. More precisely, the nilpotency of $Z_{\mathcal{ℱ}}\left(X\right)$ is bounded above by the $\mathcal{ℱ}$–killing length of $X$, a new numerical invariant which equals the number of steps it takes to make $X$ contractible by successively attaching cones on wedges of spaces in $\mathcal{ℱ}$, and this in turn is bounded above by the $\mathcal{ℱ}$–cone length of X. We then calculate or estimate the nilpotency of $Z_{\mathcal{ℱ}}\left(X\right)$ when $\mathcal{ℱ}=\mathcal{S}$, $\mathcal{ℳ}$ or $\Sigma$ for the following classes of spaces: (1) projective spaces (2) certain Lie groups such as $SU\left(n\right)$ and $Sp\left(n\right)$. The paper concludes with several open problems.