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.
Notes on file formats
Martin Arkowitz and Jeffrey Strom
Dartmouth College, Hanover NH 03755, USA
Email: martin.arkowitz@dartmouth.edu, jeffrey.strom@dartmouth.edu
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