Cup-one algebras and 1-minimal models
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| Format: | Preprint |
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2023
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| author | Porter, Richard D. Suciu, Alexander I. |
| author_facet | Porter, Richard D. Suciu, Alexander I. |
| contents | In previous work we introduced the notion of binomial cup-one algebras, which are differential graded algebras endowed with Steenrod $\cup_1$-products and compatible binomial operations. In this paper we show that binomial cup-one algebras capture homotopy 1-type. In particular, given such an $R$-dga, $(A,d_A)$, defined over the ring $R=\mathbb{Z}$ or $\mathbb{F}_p$ (for $p$ a prime), with $H^0(A)=R$ and with $H^1(A)$ a finitely generated, free $R$-module, we show that $A$ admits a functorially defined 1-minimal model, $ρ\colon (\mathcal{M}(A),d)\to (A,d_A)$, which is unique up to isomorphism. Furthermore, we associate to this model a pronilpotent group, whose continuous cohomology is isomorphic to that of $\mathcal{M}(A)$. These constructions, which refine classical notions from rational homotopy theory, allow us to distinguish spaces with isomorphic torsion-free integral cohomology rings. Moreover, we show that there is an equivalence of categories between isomorphism classes of finitely-generated, torsion-free-nilpotent groups and isomorphism classes of finitely generated 1-minimal models over the integers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_11849 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cup-one algebras and 1-minimal models Porter, Richard D. Suciu, Alexander I. Algebraic Topology Rings and Algebras 16E45, 13F20, 20F18, 20J05, 55N45, 55P62, 55S05, 55U10 In previous work we introduced the notion of binomial cup-one algebras, which are differential graded algebras endowed with Steenrod $\cup_1$-products and compatible binomial operations. In this paper we show that binomial cup-one algebras capture homotopy 1-type. In particular, given such an $R$-dga, $(A,d_A)$, defined over the ring $R=\mathbb{Z}$ or $\mathbb{F}_p$ (for $p$ a prime), with $H^0(A)=R$ and with $H^1(A)$ a finitely generated, free $R$-module, we show that $A$ admits a functorially defined 1-minimal model, $ρ\colon (\mathcal{M}(A),d)\to (A,d_A)$, which is unique up to isomorphism. Furthermore, we associate to this model a pronilpotent group, whose continuous cohomology is isomorphic to that of $\mathcal{M}(A)$. These constructions, which refine classical notions from rational homotopy theory, allow us to distinguish spaces with isomorphic torsion-free integral cohomology rings. Moreover, we show that there is an equivalence of categories between isomorphism classes of finitely-generated, torsion-free-nilpotent groups and isomorphism classes of finitely generated 1-minimal models over the integers. |
| title | Cup-one algebras and 1-minimal models |
| topic | Algebraic Topology Rings and Algebras 16E45, 13F20, 20F18, 20J05, 55N45, 55P62, 55S05, 55U10 |
| url | https://arxiv.org/abs/2306.11849 |