  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.

Algebraic Topology