Homotopy classes that are trivial mod ℱ

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

Z_{Σ} (X, Y) \subseteq Z_{ℳ} (X, Y) \subseteq Z_{§} (X, Y),

and we find examples of finite complexes $X$ and $Y$ for which these inclusions are strict. We are also interested in $Z_{ℱ} (X) = Z_{ℱ} (X, X)$ , which under composition has the structure of a semigroup with zero. We show that if $X$ is a finite dimensional complex and $ℱ = S$ , $ℳ$ or $Σ$ , then the semigroup $Z_{ℱ} (X)$ is nilpotent. More precisely, the nilpotency of $Z_{ℱ} (X)$ is bounded above by the $ℱ$ –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 $ℱ$ , and this in turn is bounded above by the $ℱ$ –cone length of X. We then calculate or estimate the nilpotency of $Z_{ℱ} (X)$ when $ℱ = S$ , $ℳ$ or $Σ$ for the following classes of spaces: (1) projective spaces (2) certain Lie groups such as $S U (n)$ and $S p (n)$ . The paper concludes with several open problems.

Keywords

cone length, trivial homotopy

Mathematical Subject Classification 2000

Primary: 55Q05

Secondary: 55P65, 55P45, 55M30

References

Forward citations

Publication

Received: 7 December 2000

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

Volume 1, issue 1 (2001)

Martin Arkowitz and Jeffrey Strom