Algebraic and Geometric Topology 1 (2001), paper no. 19, pages 381-409.

Homotopy classes that are trivial mod F

Martin Arkowitz and Jeffrey Strom

Abstract. If F is a collection of topological spaces, then a homotopy class \alpha in [X,Y] is called F-trivial if \alpha _* = 0: [A,X] --> [A,Y] 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 = \Sigma , the collection of suspensions. Clearly Z_\Sigma (X,Y) \subset Z_M(X,Y) \subset Z_S(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 \Sigma , 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 \Sigma 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

AMS subject classification. Primary: 55Q05. Secondary: 55P65, 55P45, 55M30.

DOI: 10.2140/agt.2001.1.381

E-print: arXiv:math.AT/0106184

Submitted: 7 December 2000. (Revised: 24 May 2000.) Accepted: 18 June 2001. Published: 19 June 2001.

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Martin Arkowitz and Jeffrey Strom
Dartmouth College, Hanover NH 03755, USA

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