The Negation Of Singer's Conjecture For The Sixth Algebraic Transfer
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916072880340992 |
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| author | Phuc, Dang Vo |
| author_facet | Phuc, Dang Vo |
| contents | Let $\mathscr A$ be the Steenrod algebra over the field of characteristic two, $\mathbb F_2.$ Denote by $GL(q)$ the general linear group of rank $q$ over $\mathbb F_2.$ The algebraic transfer, introduced by W. Singer [Math. Z. 202 (1989), 493-523], is a rather effective tool for unraveling the intricate structure of the (mod-2) cohomology of the Steenrod algebra, ${\rm Ext}_{\mathscr A}^{q,*}(\mathbb F_2, \mathbb F_2).$ The Kameko homomorphism is one of the useful tools to study the dimension of the domain of the Singer transfer. Singer conjectured that the algebraic transfer is always a monomorphism. This conjecture has stood for nearly 40 years, and in this work we demonstrate that it fails in general. More precisely, by developing a novel algorithm in the computer algebra system OSCAR for computing $GL(q)$-invariants of the kernel of the Kameko homomorphism, we disprove Singer's conjecture for bidegree $(6, 6+36)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09455 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Negation Of Singer's Conjecture For The Sixth Algebraic Transfer Phuc, Dang Vo Algebraic Topology Rings and Algebras Representation Theory 55Q45, 55S10, 55S05, 55T15, 68W30 Let $\mathscr A$ be the Steenrod algebra over the field of characteristic two, $\mathbb F_2.$ Denote by $GL(q)$ the general linear group of rank $q$ over $\mathbb F_2.$ The algebraic transfer, introduced by W. Singer [Math. Z. 202 (1989), 493-523], is a rather effective tool for unraveling the intricate structure of the (mod-2) cohomology of the Steenrod algebra, ${\rm Ext}_{\mathscr A}^{q,*}(\mathbb F_2, \mathbb F_2).$ The Kameko homomorphism is one of the useful tools to study the dimension of the domain of the Singer transfer. Singer conjectured that the algebraic transfer is always a monomorphism. This conjecture has stood for nearly 40 years, and in this work we demonstrate that it fails in general. More precisely, by developing a novel algorithm in the computer algebra system OSCAR for computing $GL(q)$-invariants of the kernel of the Kameko homomorphism, we disprove Singer's conjecture for bidegree $(6, 6+36)$. |
| title | The Negation Of Singer's Conjecture For The Sixth Algebraic Transfer |
| topic | Algebraic Topology Rings and Algebras Representation Theory 55Q45, 55S10, 55S05, 55T15, 68W30 |
| url | https://arxiv.org/abs/2509.09455 |