## Abstract

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

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

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

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

and we find examples of *finite complexes* $X$ and $Y$ for which these inclusions are strict. We are also interested in ${Z}_{\mathcal{\mathcal{F}}}\left(X\right)={Z}_{\mathcal{\mathcal{F}}}\left(X,X\right)$, which under composition has the structure of a semigroup with zero. We show that if $X$ is a finite dimensional complex and $\mathcal{\mathcal{F}}=\mathcal{S}$, $\mathcal{\mathcal{M}}$ or $\Sigma $, then the semigroup ${Z}_{\mathcal{\mathcal{F}}}\left(X\right)$ is nilpotent. More precisely, the nilpotency of ${Z}_{\mathcal{\mathcal{F}}}\left(X\right)$ is bounded above by the $\mathcal{\mathcal{F}}$–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 $\mathcal{\mathcal{F}}$, and this in turn is bounded above by the $\mathcal{\mathcal{F}}$–cone length of X. We then calculate or estimate the nilpotency of ${Z}_{\mathcal{\mathcal{F}}}\left(X\right)$ when $\mathcal{\mathcal{F}}=\mathcal{S}$, $\mathcal{\mathcal{M}}$ 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.

## Citation

Martin Arkowitz. Jeffrey Strom. "Homotopy classes that are trivial mod $\mathcal{F}$." Algebr. Geom. Topol. 1 (1) 381 - 409, 2001. https://doi.org/10.2140/agt.2001.1.381

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