Computational Approaches to the Singer Transfer: Preimages in the Lambda Algebra and $G_k$-Invariant Theory
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911075550625792 |
|---|---|
| author | Phuc, Dang Vo |
| author_facet | Phuc, Dang Vo |
| contents | We present a systematic, algorithmic method to compute the preimage of elements under the Singer algebraic transfer. Using the lambda algebra and the invariant-theoretic formula of P.H. Chon and L.M. Ha [5], we formulate the preimage search as a solvable problem in linear algebra. This framework is applied to study key indecomposable elements in the Adams spectral sequence. As a consequence, we show that the proof of the known result that the indecomposable element $d_0 \in \mathrm{Ext}^{4,18}_{\mathcal A}(\mathbb Z/2, \mathbb Z/2).$ lies in the image of the fourth Singer transfer, as given by Nguyen Sum in [17], is false. Furthermore, we provide the explicit description of a preimage for the indecomposable element $p_0 \in \mathrm{Ext}^{4,37}_{\mathcal A}(\mathbb Z/2, \mathbb Z/2).$ This preimage had not been explicitly determined in the previous work of N.H.V. Hung and V.T.N. Quynh [8]. Finally, our most significant contribution is the construction of a complete \textsc{SageMath} algorithm that fully automates the computation of both the dimension and an explicit basis for the $G_k$-invariant space $[(Q\mathcal{P}_k)_d]^{G_k}$. This tool facilitates the verification of our results [11, 12, 13] that were previously computed manually in connection with Singer's conjecture for rank 4. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10108 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computational Approaches to the Singer Transfer: Preimages in the Lambda Algebra and $G_k$-Invariant Theory Phuc, Dang Vo Algebraic Topology 55S05, 55S10, 55T15 We present a systematic, algorithmic method to compute the preimage of elements under the Singer algebraic transfer. Using the lambda algebra and the invariant-theoretic formula of P.H. Chon and L.M. Ha [5], we formulate the preimage search as a solvable problem in linear algebra. This framework is applied to study key indecomposable elements in the Adams spectral sequence. As a consequence, we show that the proof of the known result that the indecomposable element $d_0 \in \mathrm{Ext}^{4,18}_{\mathcal A}(\mathbb Z/2, \mathbb Z/2).$ lies in the image of the fourth Singer transfer, as given by Nguyen Sum in [17], is false. Furthermore, we provide the explicit description of a preimage for the indecomposable element $p_0 \in \mathrm{Ext}^{4,37}_{\mathcal A}(\mathbb Z/2, \mathbb Z/2).$ This preimage had not been explicitly determined in the previous work of N.H.V. Hung and V.T.N. Quynh [8]. Finally, our most significant contribution is the construction of a complete \textsc{SageMath} algorithm that fully automates the computation of both the dimension and an explicit basis for the $G_k$-invariant space $[(Q\mathcal{P}_k)_d]^{G_k}$. This tool facilitates the verification of our results [11, 12, 13] that were previously computed manually in connection with Singer's conjecture for rank 4. |
| title | Computational Approaches to the Singer Transfer: Preimages in the Lambda Algebra and $G_k$-Invariant Theory |
| topic | Algebraic Topology 55S05, 55S10, 55T15 |
| url | https://arxiv.org/abs/2507.10108 |