In this paper we give for any r, n, 0 [les ] r
[les ]
n, a Quillen's model structure to the category of simplicial
groups where
the weak equivalences are those morphisms f[bull ] such
that
πq(f[bull ]) is an isomorphism
for r
[les ] q [les ] n. This is carried out by studying the
cases
r = 0 and n → ∞ previously and, in each one
of them,
we make explicit some constructions for the associated homotopy theories,
such as
the cylinder and path objects and the loop and suspension functors, and
we also
relate the simplicial homotopy relation to the homotopy relation obtained
from these structures.