#### Volume 1, issue 1 (2001)

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

### Martin Arkowitz and Jeffrey Strom

Algebraic & Geometric Topology 1 (2001) 381–409
 arXiv: math.AT/0106184
##### Abstract

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

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

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

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

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

##### Keywords
cone length, trivial homotopy
##### Mathematical Subject Classification 2000
Primary: 55Q05
Secondary: 55P65, 55P45, 55M30
##### Publication
Revised: 24 May 2000
Accepted: 18 June 2001
Published: 19 June 2001
##### Authors
 Martin Arkowitz Dartmouth College Hanover NH 03755 USA Jeffrey Strom Dartmouth College Hanover NH 03755 USA