Abstract
If is a collection of topological spaces, then a homotopy class in is called –trivial if
for all . In this paper we study the collection of all –trivial homotopy classes in when , the collection of spheres, , the collection of Moore spaces, and , the collection of suspensions. Clearly
and we find examples of finite complexes and for which these inclusions are strict. We are also interested in , which under composition has the structure of a semigroup with zero. We show that if is a finite dimensional complex and , or , then the semigroup is nilpotent. More precisely, the nilpotency of is bounded above by the –killing length of , a new numerical invariant which equals the number of steps it takes to make contractible by successively attaching cones on wedges of spaces in , and this in turn is bounded above by the –cone length of X. We then calculate or estimate the nilpotency of when , or for the following classes of spaces: (1) projective spaces (2) certain Lie groups such as and . 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
Information