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

TL;DR

This paper develops invariants for the rational homotopy types of closed manifolds using minimal cyclic $C_ abla$-algebras, providing new proofs and extensions of known formality results based on cohomology and isotopy invariants.

Contribution

It introduces a stratification and obstruction classes for minimal $C_ abla$-algebra enhancements, leading to a complete set of invariants for rational homotopy types of certain closed manifolds.

Findings

Defines invariants for rational homotopy types using isotopy modulo (l-2)

Proves that certain high-connectivity manifolds are intrinsically formal under specific conditions

Provides new proofs and extensions of existing formality theorems by Crowley--Nordström and Cavalcanti

Abstract

Using the notion of isotopy modulo $k$, with $k \in \mathbb{N}^+$, we introduce a stratification on the set of all minimal $C_\infty$-algebra enhancements of a finite-type graded commutative algebra $H^*$. We determine obstruction classes defining the extendability of isotopy modulo $k$ to isotopy modulo $(k+1)$ for minimal $C_\infty$-algebra enhancements of $H^*$ 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 $H^*(M, \mathbb{F})$ and the isotopy modulo $(l-2)$ of the corresponding minimal unital cyclic $C_\infty$-algebra enhancements of $H^*(M, \mathbb{F})$ 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--Nordstr\"om \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 $\varphi \in H^{2r-1}(M, \mathbb{R})$ such that the map $H^r(M, \mathbb{R}) \to H^{3r-1}(M, \mathbb{R}), x \mapsto \varphi \cup x$ 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.