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

Abstract

If is a collection of topological spaces, then a homotopy class α in [X,Y] is called –trivial if

α = 0 : [ A , X ] [ A , Y ]

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

Z Σ ( X , Y ) Z ( X , Y ) 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(X)=Z(X,X), which under composition has the structure of a semigroup with zero. We show that if X is a finite dimensional complex and =S, or Σ, then the semigroup Z(X) is nilpotent. More precisely, the nilpotency of Z(X) is bounded above by the –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 , and this in turn is bounded above by the –cone length of X. We then calculate or estimate the nilpotency of Z(X) when =S, or Σ for the following classes of spaces: (1) projective spaces (2) certain Lie groups such as SU(n) and Sp(n). The paper concludes with several open problems.

Citation

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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

Received: 7 December 2000; Revised: 24 May 2000; Accepted: 18 June 2001; Published: 2001
First available in Project Euclid: 21 December 2017

zbMATH: 0982.55009
MathSciNet: MR1835263
Digital Object Identifier: 10.2140/agt.2001.1.381

Subjects:
Primary: 55Q05
Secondary: 55M30, 55P45, 55P65

Rights: Copyright © 2001 Mathematical Sciences Publishers

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