Unital $C_\infty$-algebras and the real homotopy type of $(r-1)$-connected compact manifolds of dimension $\le \ell(r-1)+2$
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866917970099306496 |
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| author | Fiorenza, Domenico Lê, Hông Vân |
| author_facet | Fiorenza, Domenico Lê, Hông Vân |
| contents | We encode the real homotopy type of an $n$-dimensional $(r-1)$-connected compact manifold $M$, $ r\ge 2$ into a minimal unital $C_\infty$-structure on $H^* (M,\mathbb R)$, obtained via homotopy transfer of the unital DGCA structure of the small quotient algebra associated with a Hodge decomposition of the de Rham algebra $\mathcal A^*(M)$, which has been proposed by Fiorenza-Kawai-Lê-Schwachhöfer in [Ann. Sc. Norm. Super Pisa (5), vol. XXII (2021), 79-107]. We prove that if $n \le \ell (r-1) +2$, with $\ell \geq 4$, the multiplication $μ_k$ on the minimal unital $C_\infty$-algebra $H^*(M,\mathbb R)$ vanishes for all $k \ge \ell-1$. This extends the results from [loc. cit.], extending the bound on the dimension from $5r-3$ to the general bound $\ell(r-1) +2$. We also prove a variant of this result, conjectured by Zhou, stating that if $n \le \ell(r-1)+4$ and $b_r (M) =1$ then the multiplication $μ_k$ for all $k \ge \ell-1$ vanishes. This implies two formality results by Cavalcanti [Math. Proc. Cambridge Philos. Soc. 141 (2006), 101-112]. We show that in any dimension $n$ the Harrison cohomology class $[μ_3]\in \mathrm {HHarr}^{3,-1}(H^* (M, \mathbb R), H^*(M, \mathbb R)) $ is a homotopy invariant of $M$ and the first obstruction to formality, and provide a detailed proof that if $n\leq 4r-1$ this is the only obstruction. Furthermore, we show that in any dimension $n$ the class $[μ_3]$ and the Bianchi-Massey tensor invented by Crowley-Nordström in [J. Topol. 13(2020), 539-575] define each other uniquely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_19506 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Unital $C_\infty$-algebras and the real homotopy type of $(r-1)$-connected compact manifolds of dimension $\le \ell(r-1)+2$ Fiorenza, Domenico Lê, Hông Vân Algebraic Topology Commutative Algebra Differential Geometry K-Theory and Homology 55P62, 57R19, 13D03, 58A10 We encode the real homotopy type of an $n$-dimensional $(r-1)$-connected compact manifold $M$, $ r\ge 2$ into a minimal unital $C_\infty$-structure on $H^* (M,\mathbb R)$, obtained via homotopy transfer of the unital DGCA structure of the small quotient algebra associated with a Hodge decomposition of the de Rham algebra $\mathcal A^*(M)$, which has been proposed by Fiorenza-Kawai-Lê-Schwachhöfer in [Ann. Sc. Norm. Super Pisa (5), vol. XXII (2021), 79-107]. We prove that if $n \le \ell (r-1) +2$, with $\ell \geq 4$, the multiplication $μ_k$ on the minimal unital $C_\infty$-algebra $H^*(M,\mathbb R)$ vanishes for all $k \ge \ell-1$. This extends the results from [loc. cit.], extending the bound on the dimension from $5r-3$ to the general bound $\ell(r-1) +2$. We also prove a variant of this result, conjectured by Zhou, stating that if $n \le \ell(r-1)+4$ and $b_r (M) =1$ then the multiplication $μ_k$ for all $k \ge \ell-1$ vanishes. This implies two formality results by Cavalcanti [Math. Proc. Cambridge Philos. Soc. 141 (2006), 101-112]. We show that in any dimension $n$ the Harrison cohomology class $[μ_3]\in \mathrm {HHarr}^{3,-1}(H^* (M, \mathbb R), H^*(M, \mathbb R)) $ is a homotopy invariant of $M$ and the first obstruction to formality, and provide a detailed proof that if $n\leq 4r-1$ this is the only obstruction. Furthermore, we show that in any dimension $n$ the class $[μ_3]$ and the Bianchi-Massey tensor invented by Crowley-Nordström in [J. Topol. 13(2020), 539-575] define each other uniquely. |
| title | Unital $C_\infty$-algebras and the real homotopy type of $(r-1)$-connected compact manifolds of dimension $\le \ell(r-1)+2$ |
| topic | Algebraic Topology Commutative Algebra Differential Geometry K-Theory and Homology 55P62, 57R19, 13D03, 58A10 |
| url | https://arxiv.org/abs/2310.19506 |