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

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 $H^{\ast }(M,\mathbb{R})$, obtained via homotopy transfer of the unital DGCA structure of the small quotient algebra associated with a Hodge decomposition of the de Rham algebra ${\mathcal{A}}^{\ast }(M)$, 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 $\ell \ge 4$, the multiplication ${\mu }_k$ on the minimal unital $C_{\infty }$-algebra $H^{\ast }(M,\mathbb{R})$ 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 ${\mu }_k$ 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 $[{\mu }_3]\in {\mathrm{HHarr}}^{3,-1}(H^{\ast }(M,\mathbb{R}),H^{\ast }(M,\mathbb{R}))$ is a homotopy invariant of $M$ and the first obstruction to formality, and provide a detailed proof that if $n\le 4r-1$ this is the only obstruction. Furthermore, we show that in any dimension $n$ the class $[{\mu }_3]$ and the Bianchi-Massey tensor invented by Crowley-Nordström in [J. Topol. 13(2020), 539-575] define each other uniquely.

v6, 32 p.; affiliation added, a reference added, few typos corrected. Final version. To appear in Ann. Sc. Norm. Super. Pisa