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 $H^*$. 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 $n\leq \ell(r-1)+2,\, r\geq2,\, \ell\geq 4$, then its real and rational homotopy types are determined by its cohomology algebra $H^*(M;\mathbb F)$ 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--Nordstr\"om \cite{CN}: if $M$ is a closed $(r-1)$-connected manifold of dimension $4r-1$ with $b_r(M)\leq3$, and there exists a class $\varphi\in H^{2r-1}(M;\mathbb R)$ such that multiplication by $\varphi$ induces an isomorphism $H^r(M;\mathbb R)\longrightarrow H^{3r-1}(M;\mathbb R),\,x\longmapsto\varphi\smile x$, then $M$ is intrinsically formal. Finally, we prove a borderline extension of a vanishing theorem of Fiorenza--L\^e: if an $(r-1)$-connected Poincar\'e DGCA over $\mathbb Q$ of degree $n\leq5r-2$ admits a Hodge homotopy and satisfies $b^r\leq2$, then the operations of its transferred minimal unital cyclic $C_\infty$-algebra vanish in every arity $k\geq4$.