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

Abstract

Using the notion of isotopy modulo $k$, for $k\in {\mathbb{N}}^{+}$, we introduce a stratification on the set of minimal $C_{\infty }$-algebra enhancements of a finite-dimensional graded commutative algebra $H^{\ast }$. We prove that two such enhancements are $C_{\infty }$-isotopic if and only if they are isotopic modulo $k$ for every $k\in {\mathbb{N}}^{+}$. We define obstruction sets governing the extension of an isotopy modulo $k$ to an isotopy modulo $(k+1)$ and establish their generalized additivity. We prove that if $M$ is a closed $(r-1)$-connected manifold of dimension $n\le \ell (r-1)+2,r\ge 2,\ell \ge 4$, then its real and rational homotopy types are determined by its cohomology algebra $H^{\ast }(M;\mathbb{F})$ together with the isotopy class modulo $(\ell -2)$ of the corresponding minimal unital cyclic $C_{\infty }$-algebra enhancement, for $\mathbb{F}=\mathbb{R}$ and $\mathbb{F}=\mathbb{Q}$, respectively. Combining this obstruction theory with the Hodge homotopy method introduced in \cite{FKLS2021} and further developed in \cite{FiorenzaLe2025}, we give a new proof of a theorem of Crowley--Nordström \cite{CN}: if $M$ is a closed $(r-1)$-connected manifold of dimension $4r-1$ with $b_r(M)\le 3$, and there exists a class $\phi \in H^{2r-1}(M;\mathbb{R})$ such that multiplication by $\phi$ induces an isomorphism $H^r(M;\mathbb{R})⟶H^{3r-1}(M;\mathbb{R}),x⟼\phi ⌣x$, then $M$ is intrinsically formal. Finally, we prove a borderline extension of a vanishing theorem of Fiorenza--Lê: if an $(r-1)$-connected Poincaré DGCA over $\mathbb{Q}$ of degree $n\le 5r-2$ admits a Hodge homotopy and satisfies $b^r\le 2$, then the operations of its transferred minimal unital cyclic $C_{\infty }$-algebra vanish in every arity $k\ge 4$.

v4:45 p. Minor corrections and improvements. Section 5 was written in collaboration with Domenico Fiorenza