Let
T
r
k
:
F
2
⊗
G
L
k
P
H
i
(
B
V
k
)
→
E
x
t
A
k
,
k
+
i
(
F
2
,
F
2
)
Tr_k:\mathbb {F}_2\underset {GL_k}{\otimes } PH_i(B\mathbb {V}_k)\to Ext_{\mathcal {A}}^{k,k+i}(\mathbb {F}_2, \mathbb {F}_2)
be the algebraic transfer, which is defined by W. Singer as an algebraic version of the geometrical transfer
t
r
k
:
π
∗
S
(
(
B
V
k
)
+
)
→
π
∗
S
(
S
0
)
tr_k: \pi _*^S((B\mathbb {V} _k)_+) \to \pi _*^S(S^0)
. It has been shown that the algebraic transfer is highly nontrivial and, more precisely, that
T
r
k
Tr_k
is an isomorphism for
k
=
1
,
2
,
3
k=1, 2, 3
. However, Singer showed that
T
r
5
Tr_5
is not an epimorphism. In this paper, we prove that
T
r
4
Tr_4
does not detect the nonzero element
g
s
∈
E
x
t
A
4
,
12
⋅
2
s
(
F
2
,
F
2
)
g_s\in Ext_{\mathcal {A}}^{4,12\cdot 2^s}(\mathbb {F}_2, \mathbb {F}_2)
for every
s
≥
1
s\geq 1
. As a consequence, the localized
(
S
q
0
)
−
1
T
r
4
(Sq^0)^{-1}Tr_4
given by inverting the squaring operation
S
q
0
Sq^0
is not an epimorphism. This gives a negative answer to a prediction by Minami.