Unital $C_\infty$-algebras and the real homotopy type of $(r-1)$-connected compact manifolds of dimension $\le \ell(r-1)+2$

TL;DR

This work develops a unital $C_\infty$-algebra framework, via a Hodge homotopy transfer from Fiorenza–Kawai–Lê–Schwachenhöfer’s small quotient DGCA, to encode the real homotopy type of $(r-1)$-connected compact manifolds of dimension $n$ in terms of a minimal unital $C_\infty$-structure on $H^*(M,\mathbb{R})$. It proves that for $n\le\ell(r-1)+2$ with $\ell\ge4$, all higher multiplications $\mu_k$ vanish for $k\ge\ell-1$, extending FKLS2021 from $5r-3$ to the general bound, and establishes a Zhou-type variant under $n\le\ell(r-1)+4$ and $b_r(M)=1$, implying Cavalcanti formality results. The Harrison cohomology class $[\mu_3]$ is shown to be a homotopy invariant and the first obstruction to formality, and, in the range $n\le 4r-1$, it completely determines the DGCA-homotopy type; moreover, $[\mu_3]$ and the Bianchi-Massey tensor define each other uniquely, linking obstruction theory to Crowley–Nordström’s tensor. Collectively, these results yield a concrete, computable invariant framework for the real homotopy type of a broad class of manifolds, with clear connections to formality questions and existing invariants. The approach sharpens dimension bounds and unifies several formality criteria in a homotopy-coherent, $C_\infty$-algebraic setting.

Abstract

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.

Theorems & Definitions (48)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Remark 2.6
• Example 2.7
• Remark 2.8
• Definition 2.9
• Proposition 2.10