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On A-generators of the cohomology H∗(V⨁5)=Z/2[u1,…,u5] and the cohomological transfer of rank 5
Let denote V ⊕ n the n -dimensional vector space over the prime field Z / 2 . We write A as the 2-primary Steenrod algebra, which is the algebra of stable natural endomorphisms of the mod 2 cohomology functor on topological spaces. Working at the prime 2, computing the cohomology of A is an importan...
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Published in: | Rendiconti del Circolo matematico di Palermo 2024-04, Vol.73 (3), p.989-1007 |
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Main Author: | |
Format: | Article |
Language: | English |
Subjects: | |
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Summary: | Let denote
V
⊕
n
the
n
-dimensional vector space over the prime field
Z
/
2
.
We write
A
as the 2-primary Steenrod algebra, which is the algebra of stable natural endomorphisms of the mod 2 cohomology functor on topological spaces. Working at the prime 2, computing the cohomology of
A
is an important problem of Algebraic topology since it is the initial page of the Adams spectral sequence (the ASS) converging to stable homotopy groups of the spheres. A particularly effective technique for characterizing this cohomology is the cohomological transfer of rank
n
, which was first introduced by W. Singer in his seminal work [Math. Z. 202, 493–523 (1989)]. This transfer maps from a certain subquotient of a divided power algebra to the cohomology of
A
. Actually, the Singer transfer is induced over the
E
2
-term of the ASS by the geometrical transfer map
Σ
∞
(
B
(
V
⊕
n
)
+
)
⟶
Σ
∞
(
S
0
)
in stable homotopy theory. Singer formulated a pivotal conjecture that
the cohomological transfer is always a one-to-one homomorphism
, but its validity remains unknown for any
n
≥
5
.
The Singer transfer is closely linked to the classical “hit problem” first proposed by Frank Peterson in [Abstracts Papers Presented Am. Math. Soc. 833, 55–89 (1987)]. The hit problem involves finding a minimal generating set for the unstable
A
-module
H
∗
(
V
⊕
n
)
=
Z
/
2
[
u
1
,
…
,
u
n
]
.
Despite several decades of research, this problem remains unsolved for all
n
≥
5
. In this paper, we study the hit problem for the
A
-module
H
∗
(
V
⊕
n
)
and verify Singer’s conjecture for the cases where
n
=
5
and the general degree
d
=
2
t
+
5
+
2
t
+
2
+
2
t
+
1
-
5
for any non-negative integer
t
. The results of our study demonstrate that the Singer cohomological transfer is an isomorphism for
n
=
5
in degree
d
. This provides a positive answer to Singer’s conjecture in the considered cases. |
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ISSN: | 0009-725X 1973-4409 |
DOI: | 10.1007/s12215-023-00964-7 |