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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
Main Author: Phúc, Ɖặng Võ
Format: Article
Language:English
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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.
ISSN:0009-725X
1973-4409
DOI:10.1007/s12215-023-00964-7