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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 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

ZΣ(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.

Keywords
cone length, trivial homotopy
Mathematical Subject Classification 2000
Primary: 55Q05
Secondary: 55P65, 55P45, 55M30
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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