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If
is a collection of topological spaces, then a homotopy class
in
is called
–trivial
if
for all . In this paper
we study the collection
of all –trivial homotopy
classes in when
, the collection of
spheres, , the collection
of Moore spaces, and ,
the collection of suspensions. Clearly
and we find examples of finite complexes
and
for which these inclusions are strict. We are also interested in
,
which under composition has the structure of a semigroup with zero. We show that if
is a finite dimensional
complex and ,
or
, then the
semigroup
is nilpotent. More precisely, the nilpotency of
is bounded above
by the –killing
length of ,
a new numerical invariant which equals the number of steps it takes to make
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
when
,
or
for
the following classes of spaces: (1) projective spaces (2) certain Lie groups such as
and
. The
paper concludes with several open problems.
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