Let
be a pointed CW–complex. The generalized conjecture on
spherical classes states that
the Hurewicz homomorphismvanishes on classesof of Adams filtrationgreater than .
Let
denote
the
Lannes–Zarati homomorphism for the unstable
–module
. When
,
this homomorphism corresponds to an associated graded of the
Hurewicz map. An algebraic version of the conjecture states that the
Lannes–Zarati
homomorphism,
, vanishes
in any positive stem for
and
for any unstable
–module
.
We prove that, for
an
unstable
–module of finite
type, the
Lannes–Zarati
homomorphism,
,
vanishes on decomposable elements of the form
in positive
stems, where
and
with
either
,
and
, or
,
and
.
Consequently, we obtain a theorem proved by Hưng and Peterson in
1998. We also prove that the fifth Lannes–Zarati homomorphism for
vanishes on decomposable elements in positive stems.
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