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If F is a collection of topological spaces, then a
homotopy class α in [X,Y] is called F–trivial if
![α<sub>*</sub> = 0: [A,X] → [A,Y]](a190x.png)
for all A in F. In this paper we study the collection
ZF(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 = Σ, the collection of suspensions. Clearly
and we find examples of finite complexes
X and Y for which these inclusions are strict. We are also interested
in ZF(X) = ZF(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 Σ, then the semigroup
ZF(X) is nilpotent. More precisely, the nilpotency of
ZF(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 ZF(X) when F =
S, M 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.
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