Minimal Unital Cyclic $C_{\mathrm{\infty }}$-Algebras and the Real and Rational Homotopy Type of Closed Manifolds

Using the notion of isotopy modulo $k$, with $k\in {\mathbb{N}}^{+}$, we introduce a stratification on the set of all minimal $C_{\mathrm{\infty }}$-algebra enhancements of a finite-type graded commutative algebra $H^{\ast }$. We determine obstruction classes defining the extendability of isotopy modulo $k$ to isotopy modulo $(k+1)$ for minimal $C_{\mathrm{\infty }}$-algebra enhancements of $H^{\ast }$ and demonstrate their generalized additivity. As a result, we define a complete set of invariants of the rational homotopy types of closed simply connected manifolds $M$. We prove that if $M$ is a closed $(r-1)$-connected manifold of dimension $n\le l(r-1)+2$ (where $r\ge 2,l\ge 4$), the real and rational homotopy type of $M$ is defined uniquely by the cohomology algebra $H^{\ast }(M,\mathbb{F})$ and the isotopy modulo $(l-2)$ of the corresponding minimal unital cyclic $C_{\mathrm{\infty }}$-algebra enhancements of $H^{\ast }(M,\mathbb{F})$ for $\mathbb{F}=\mathbb{R},\mathbb{Q}$, respectively. Combining this with the Hodge homotopy introduced in \cite{FKLS2021} and developed in \cite{FiorenzaLe2025}, we provide a new proof of a theorem by Crowley–Nordström \cite{CN}: a $(r-1)$-connected closed manifold $M$ of dimension $4r-1$ with $b_r(M)\le 3$ is intrinsically formal if there exists a $\phi \in H^{2r-1}(M,\mathbb{R})$ such that the map $H^r(M,\mathbb{R})\to H^{3r-1}(M,\mathbb{R}),x\mapsto \phi \cup x$ is an isomorphism. Furthermore, we provide a new proof and extension of Cavalcanti’s result \cite{Cavalcanti2006}, showing that a $(r-1)$-connected closed manifold $M$ of dimension $4r$ with $b_r(M)\le 2$ is intrinsically formal under similar conditions.