A counter-example to Singer's conjecture for the algebraic transfer
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914923470127104 |
|---|---|
| author | Sum, Nguyen |
| author_facet | Sum, Nguyen |
| contents | Write $P_k:= \mathbb F_2[x_1,x_2,\ldots ,x_k]$ for the polynomial algebra over the prime field $\mathbb F_2$ with two elements, in $k$ generators $x_1, x_2, \ldots , x_k$, each of degree 1. The polynomial algebra $P_k$ is considered as a module over the mod-2 Steenrod algebra, $\mathcal A$. Let $GL_k$ be the general linear group over the field $\mathbb F_2$. This group acts naturally on $P_k$ by matrix substitution. Since the two actions of $\mathcal A$ and $GL_k$ upon $P_k$ commute with each other, there is an inherit action of $GL_k$ on $\mathbb F_2{\otimes}_{\mathcal A}P_k$. Denote by $(\mathbb F_2{\otimes}_{\mathcal A}P_k)_n^{GL_k}$ the subspace of $\mathbb F_2{\otimes}_{\mathcal A}P_k$ consisting of all the $GL_k$-invariant classes of degree $n$. In 1989, Singer [24] defined the homological algebraic transfer $$φ_k :\mbox{Tor}^{\mathcal A}_{k,k+n}(\mathbb F_2,\mathbb F_2) \longrightarrow (\mathbb F_2{\otimes}_{\mathcal A}P_k)_n^{GL_k},$$ where $\mbox{Tor}^{\mathcal{A}}_{k, k+n}(\mathbb{F}_2, \mathbb{F}_2)$ is the dual of Ext$_{\mathcal{A}}^{k,k+n}(\mathbb F_2,\mathbb F_2)$, the $E_2$ term of the Adams spectral sequence of spheres. In general, the transfer $φ_k$ is not a monomorphism and Singer made a conjecture that $φ_k$ is an epimorphism for any $k \geqslant 0$. The conjecture is studied by many authors. It is true for $k \leqslant 3$ but unknown for $k \geqslant 4$. In this paper, by using a technique of the Peterson hit problem we prove that Singer's conjecture is not true for $k=5$ and the internal degree $n = 108$. This result also refutes a one of Phúc in [19]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06669 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A counter-example to Singer's conjecture for the algebraic transfer Sum, Nguyen Algebraic Topology 55S10, 55S05, 55T15 Write $P_k:= \mathbb F_2[x_1,x_2,\ldots ,x_k]$ for the polynomial algebra over the prime field $\mathbb F_2$ with two elements, in $k$ generators $x_1, x_2, \ldots , x_k$, each of degree 1. The polynomial algebra $P_k$ is considered as a module over the mod-2 Steenrod algebra, $\mathcal A$. Let $GL_k$ be the general linear group over the field $\mathbb F_2$. This group acts naturally on $P_k$ by matrix substitution. Since the two actions of $\mathcal A$ and $GL_k$ upon $P_k$ commute with each other, there is an inherit action of $GL_k$ on $\mathbb F_2{\otimes}_{\mathcal A}P_k$. Denote by $(\mathbb F_2{\otimes}_{\mathcal A}P_k)_n^{GL_k}$ the subspace of $\mathbb F_2{\otimes}_{\mathcal A}P_k$ consisting of all the $GL_k$-invariant classes of degree $n$. In 1989, Singer [24] defined the homological algebraic transfer $$φ_k :\mbox{Tor}^{\mathcal A}_{k,k+n}(\mathbb F_2,\mathbb F_2) \longrightarrow (\mathbb F_2{\otimes}_{\mathcal A}P_k)_n^{GL_k},$$ where $\mbox{Tor}^{\mathcal{A}}_{k, k+n}(\mathbb{F}_2, \mathbb{F}_2)$ is the dual of Ext$_{\mathcal{A}}^{k,k+n}(\mathbb F_2,\mathbb F_2)$, the $E_2$ term of the Adams spectral sequence of spheres. In general, the transfer $φ_k$ is not a monomorphism and Singer made a conjecture that $φ_k$ is an epimorphism for any $k \geqslant 0$. The conjecture is studied by many authors. It is true for $k \leqslant 3$ but unknown for $k \geqslant 4$. In this paper, by using a technique of the Peterson hit problem we prove that Singer's conjecture is not true for $k=5$ and the internal degree $n = 108$. This result also refutes a one of Phúc in [19]. |
| title | A counter-example to Singer's conjecture for the algebraic transfer |
| topic | Algebraic Topology 55S10, 55S05, 55T15 |
| url | https://arxiv.org/abs/2408.06669 |