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Abstract
Let
X
be a pointed CW–complex. The generalized conjecture on
spherical classes states that
the Hurewicz homomorphism
H
: π ∗ ( Q 0 X )
→ H ∗ ( Q 0 X )
vanishes on classes
of π ∗ ( Q 0 X )
of Adams filtration
greater than 2 .
Let
φ s M : Ext A s ( M , F 2 )
→ ( F 2 ⊗ A R s M ) ∗ denote
the
s th
Lannes–Zarati homomorphism for the unstable
A –module
M . When
M
=
H ̃ ∗ ( X ) ,
this homomorphism corresponds to an associated graded of the
Hurewicz map. An algebraic version of the conjecture states that the
s th Lannes–Zarati
homomorphism,
φ s M , vanishes
in any positive stem for
s
> 2 and
for any unstable
A –module
M .
We prove that, for
M an
unstable
A –module of finite
type, the
s th Lannes–Zarati
homomorphism,
φ s M ,
vanishes on decomposable elements of the form
α β in positive
stems, where
α
∈ Ext A p ( F 2 , F 2 )
and
β
∈ Ext A q ( M , F 2 ) with
either
p
≥ 2 ,
q
> 0 and
p
+
q
=
s , or
p
=
s
≥ 2 ,
q
= 0 and
stem ( β )
>
s
− 2 .
Consequently, we obtain a theorem proved by Hư ng and Peterson in
1998. We also prove that the fifth Lannes–Zarati homomorphism for
H ̃ ∗ ( ℝ ℙ ∞ )
vanishes on decomposable elements in positive stems.
Keywords
spherical classes, loop spaces, Adams spectral sequences,
Steenrod algebra, lambda algebra, invariant theory, Dickson
algebra
Mathematical Subject Classification 2010
Primary: 55P47, 55Q45, 55S10, 55T15
Publication
Received: 2 May 2018
Revised: 19 September 2018
Accepted: 15 October 2018
Published: 21 May 2019