The definable content of homological invariants I: $\mathrm{Ext}$ & $\mathrm{lim}^1$
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912023039705088 |
|---|---|
| author | Bergfalk, Jeffrey Lupini, Martino Panagiotopoulos, Aristotelis |
| author_facet | Bergfalk, Jeffrey Lupini, Martino Panagiotopoulos, Aristotelis |
| contents | This is the first installment in a series of papers in which we illustrate how classical invariants of homological algebra and algebraic topology can be enriched with additional descriptive set-theoretic information. To effect this enrichment, we show that many of these invariants can be naturally regarded as functors to the category, introduced herein, of groups with a Polish cover. The resulting definable invariants provide far stronger means of classification.
In the present work we focus on the first derived functors of $\mathrm{Hom}(-,-)$ and $\mathrm{lim}(-)$. The resulting definable $\mathrm{Ext}(B,F)$ for pairs of countable abelian groups $B,F$ and definable $\mathrm{lim}^{1}(\boldsymbol{A})$ for towers $\boldsymbol{A}$ of Polish abelian groups substantially refine their classical counterparts. We show, for example, that the definable $\textrm{Ext}(-,\mathbb{Z})$ is a fully faithful contravariant functor from the category of finite rank torsion-free abelian groups $Λ$ with no free summands; this contrasts with the fact that there are uncountably many non-isomorphic such groups $Λ$ with isomorphic classical invariants $\textrm{Ext}(Λ,\mathbb{Z}) $. To facilitate our analysis, we introduce a general Ulam stability framework for groups with a Polish cover and we prove several rigidity results for non-Archimedean abelian groups with a Polish cover. A special case of our main result answers a question of Kanovei and Reeken regarding quotients of the $p$-adic groups. Finally, using cocycle superrigidity methods for profinite actions of property (T) groups, we obtain a hierarchy of complexity degrees for the problem $\mathcal{R}(\mathrm{Aut}(Λ)\curvearrowright\mathrm{Ext}(Λ,\mathbb{Z}))$ of classifying all group extensions of $Λ$ by $\mathbb{Z}$ up to base-free isomorphism, when $Λ=\mathbb{Z}[1/p]^{d}$ for prime numbers $p$ and $ d\geq 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_08782 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The definable content of homological invariants I: $\mathrm{Ext}$ & $\mathrm{lim}^1$ Bergfalk, Jeffrey Lupini, Martino Panagiotopoulos, Aristotelis Logic Algebraic Topology Dynamical Systems K-Theory and Homology Primary 54H05, 18G10, 18G60, Secondary 55N05, 55N07, 37A20 This is the first installment in a series of papers in which we illustrate how classical invariants of homological algebra and algebraic topology can be enriched with additional descriptive set-theoretic information. To effect this enrichment, we show that many of these invariants can be naturally regarded as functors to the category, introduced herein, of groups with a Polish cover. The resulting definable invariants provide far stronger means of classification. In the present work we focus on the first derived functors of $\mathrm{Hom}(-,-)$ and $\mathrm{lim}(-)$. The resulting definable $\mathrm{Ext}(B,F)$ for pairs of countable abelian groups $B,F$ and definable $\mathrm{lim}^{1}(\boldsymbol{A})$ for towers $\boldsymbol{A}$ of Polish abelian groups substantially refine their classical counterparts. We show, for example, that the definable $\textrm{Ext}(-,\mathbb{Z})$ is a fully faithful contravariant functor from the category of finite rank torsion-free abelian groups $Λ$ with no free summands; this contrasts with the fact that there are uncountably many non-isomorphic such groups $Λ$ with isomorphic classical invariants $\textrm{Ext}(Λ,\mathbb{Z}) $. To facilitate our analysis, we introduce a general Ulam stability framework for groups with a Polish cover and we prove several rigidity results for non-Archimedean abelian groups with a Polish cover. A special case of our main result answers a question of Kanovei and Reeken regarding quotients of the $p$-adic groups. Finally, using cocycle superrigidity methods for profinite actions of property (T) groups, we obtain a hierarchy of complexity degrees for the problem $\mathcal{R}(\mathrm{Aut}(Λ)\curvearrowright\mathrm{Ext}(Λ,\mathbb{Z}))$ of classifying all group extensions of $Λ$ by $\mathbb{Z}$ up to base-free isomorphism, when $Λ=\mathbb{Z}[1/p]^{d}$ for prime numbers $p$ and $ d\geq 1$. |
| title | The definable content of homological invariants I: $\mathrm{Ext}$ & $\mathrm{lim}^1$ |
| topic | Logic Algebraic Topology Dynamical Systems K-Theory and Homology Primary 54H05, 18G10, 18G60, Secondary 55N05, 55N07, 37A20 |
| url | https://arxiv.org/abs/2008.08782 |