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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2508.20629 |
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| _version_ | 1866908862636883968 |
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| author | Randal-Williams, Oscar |
| author_facet | Randal-Williams, Oscar |
| contents | We propose a way to organise the subject of ``higher-order homological stability'', in the context of a graded $E_2$-algebra $\mathbf{R}$, along the same lines that the chromatic perspective organises stable homotopy theory.
From this point of view proving a (higher-order) homological stability theorem corresponds to producing Smith--Toda complexes in the category of $\mathbf{R}$-modules: using this perspective we prove that whenever $\mathbf{R}$ is defined over a field of positive characteristic and satisfies some standard properties, there is a sequence of higher-order homological stability theorems whose slopes tend to 1.
We propose that in a higher-order stable range the ``stable homology'' should be interpreted as certain Bousfield localisations in the category of $\mathbf{R}$-modules, leading to a chromatic tower and monochromatic layers. Given the existence of suitable Smith--Toda complexes we establish several properties of these localisations, in particular explaining how higher-order stabilisation maps yield periodic families in the monochromatic layers.
We explain how to associate to such an $\mathbf{R}$ a Hopf algebra which completely governs the kinds of higher-order stability maps that it enjoys, in the sense that the cohomology of this Hopf algebra has precisely the same stability patterns as $\mathbf{R}$. When $\mathbf{R}$ comes from a sequence of groups, this Hopf algebra has a concrete description as the coinvariants of the $E_1$-Steinberg modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20629 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A chromatic approach to homological stability Randal-Williams, Oscar Algebraic Topology We propose a way to organise the subject of ``higher-order homological stability'', in the context of a graded $E_2$-algebra $\mathbf{R}$, along the same lines that the chromatic perspective organises stable homotopy theory. From this point of view proving a (higher-order) homological stability theorem corresponds to producing Smith--Toda complexes in the category of $\mathbf{R}$-modules: using this perspective we prove that whenever $\mathbf{R}$ is defined over a field of positive characteristic and satisfies some standard properties, there is a sequence of higher-order homological stability theorems whose slopes tend to 1. We propose that in a higher-order stable range the ``stable homology'' should be interpreted as certain Bousfield localisations in the category of $\mathbf{R}$-modules, leading to a chromatic tower and monochromatic layers. Given the existence of suitable Smith--Toda complexes we establish several properties of these localisations, in particular explaining how higher-order stabilisation maps yield periodic families in the monochromatic layers. We explain how to associate to such an $\mathbf{R}$ a Hopf algebra which completely governs the kinds of higher-order stability maps that it enjoys, in the sense that the cohomology of this Hopf algebra has precisely the same stability patterns as $\mathbf{R}$. When $\mathbf{R}$ comes from a sequence of groups, this Hopf algebra has a concrete description as the coinvariants of the $E_1$-Steinberg modules. |
| title | A chromatic approach to homological stability |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2508.20629 |