Abstract: We introduce an equivalence relation between
idempotents of a given category. In particular, an idempotent splits if and only
if it is equivalent to an identity. In the case of finitely generated free
modules it leads to a natural group structure equivalent to the reduced
projective class group Thus each idempotent f
has a natural obstruction to be stably equivalent to an
identity idempotent. In the case of homotopy category of pointed, finite, and
connected CW complexes, each idempotent f has
an obstruction to split (see the work of Lück and
Ranicki [7]). It is shown that a generic idempotent is an idempotent on an n-dimensional
CW complex X
so that In that case is equal to where is the induced idempotent of
cellular n-chains of the universal
cover of X.
Keywords and phrases: finitely dominated CW complexes, idempotents, splitting idempotents.
