Secondary Ext of the Fiber of $Sq^n$ and the Secondary Adams Spectral Sequence
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| Formato: | Preprint |
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2026
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| _version_ | 1866910234520322048 |
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| author | Phuc, Dang Vo |
| author_facet | Phuc, Dang Vo |
| contents | At the prime $2$, let $\mathcal{B}$ denote the secondary Steenrod algebra in the sense of Baues, Baues--Jibladze, Nassau, and Baues--Frankland. We determine the secondary Ext groups of the secondary cohomology objects of the three fibers considered by Bruner in his work on the fiber of $\mathrm{Sq}^n$. For $$F_n = \mathrm{fib}(H \xrightarrow{\mathrm{Sq}^n} Σ^n H),\,\, \text{and}\,\, F_{n\mathbb{Z}} = \mathrm{fib}(H\mathbb{Z} \xrightarrow{\mathrm{Sq}^n} Σ^n H),$$ where $n>0$ in the first case and $n \ge 2$ in the second, and for $$F = \mathrm{fib}(H\mathbb{Z} \to \prod_{i>0} Σ^{2i} H),$$ where the $i$-th component is represented after mod-$2$ reduction by $\mathrm{Sq}^{2i}$, the groups are $$\mathrm{Ext}_{\mathcal{B}}^{*,*}(H^*_{\mathcal{B}} F_n, \mathbb{F}_2) \cong \mathbb{F}_2 \oplus Σ^{1,n}\mathbb{F}_2,$$ $$\mathrm{Ext}_{\mathcal{B}}^{*,*}(H^*_{\mathcal{B}} F_{n\mathbb{Z}}, \mathbb{F}_2) \cong \mathbb{F}_2[h_0] \oplus Σ^{1,n}\mathbb{F}_2,$$ and $$\mathrm{Ext}_{\mathcal{B}}^{*,*}(H^*_{\mathcal{B}} F, \mathbb{F}_2) \cong \mathbb{F}_2[h_0] \oplus \bigoplus_{j>0} Σ^{1,2^j}\mathbb{F}_2 \oplus \bigoplus_{i>0, i \neq 2^k} Σ^{0,2i-1}\mathbb{F}_2$$ (where the last direct sum runs over $i$ not a power of $2$). Here $h_0$ has bidegree $(1,1)$. The group-level calculation follows from Bruner's $E_3$-calculation and the Baues--Jibladze comparison theorem, and the proof gives an explicit primary calculation of the kernels and cokernels of the maps $D_n$, $D_{n\mathbb{Z}}$, and $D_F$. Separately, under an explicit track-fiber realization hypothesis, the same groups are computed from displayed secondary mapping fibers. Under the same hypothesis, the Bruner--Rognes Yoneda description of the ordinary Adams $d_2$ is the primary shadow of the first differential in secondary homological algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_18906 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Secondary Ext of the Fiber of $Sq^n$ and the Secondary Adams Spectral Sequence Phuc, Dang Vo Algebraic Topology Primary 55T15, 55S20, Secondary 55S10, 55P42 At the prime $2$, let $\mathcal{B}$ denote the secondary Steenrod algebra in the sense of Baues, Baues--Jibladze, Nassau, and Baues--Frankland. We determine the secondary Ext groups of the secondary cohomology objects of the three fibers considered by Bruner in his work on the fiber of $\mathrm{Sq}^n$. For $$F_n = \mathrm{fib}(H \xrightarrow{\mathrm{Sq}^n} Σ^n H),\,\, \text{and}\,\, F_{n\mathbb{Z}} = \mathrm{fib}(H\mathbb{Z} \xrightarrow{\mathrm{Sq}^n} Σ^n H),$$ where $n>0$ in the first case and $n \ge 2$ in the second, and for $$F = \mathrm{fib}(H\mathbb{Z} \to \prod_{i>0} Σ^{2i} H),$$ where the $i$-th component is represented after mod-$2$ reduction by $\mathrm{Sq}^{2i}$, the groups are $$\mathrm{Ext}_{\mathcal{B}}^{*,*}(H^*_{\mathcal{B}} F_n, \mathbb{F}_2) \cong \mathbb{F}_2 \oplus Σ^{1,n}\mathbb{F}_2,$$ $$\mathrm{Ext}_{\mathcal{B}}^{*,*}(H^*_{\mathcal{B}} F_{n\mathbb{Z}}, \mathbb{F}_2) \cong \mathbb{F}_2[h_0] \oplus Σ^{1,n}\mathbb{F}_2,$$ and $$\mathrm{Ext}_{\mathcal{B}}^{*,*}(H^*_{\mathcal{B}} F, \mathbb{F}_2) \cong \mathbb{F}_2[h_0] \oplus \bigoplus_{j>0} Σ^{1,2^j}\mathbb{F}_2 \oplus \bigoplus_{i>0, i \neq 2^k} Σ^{0,2i-1}\mathbb{F}_2$$ (where the last direct sum runs over $i$ not a power of $2$). Here $h_0$ has bidegree $(1,1)$. The group-level calculation follows from Bruner's $E_3$-calculation and the Baues--Jibladze comparison theorem, and the proof gives an explicit primary calculation of the kernels and cokernels of the maps $D_n$, $D_{n\mathbb{Z}}$, and $D_F$. Separately, under an explicit track-fiber realization hypothesis, the same groups are computed from displayed secondary mapping fibers. Under the same hypothesis, the Bruner--Rognes Yoneda description of the ordinary Adams $d_2$ is the primary shadow of the first differential in secondary homological algebra. |
| title | Secondary Ext of the Fiber of $Sq^n$ and the Secondary Adams Spectral Sequence |
| topic | Algebraic Topology Primary 55T15, 55S20, Secondary 55S10, 55P42 |
| url | https://arxiv.org/abs/2605.18906 |