Minimal Unital Cyclic C∞-Algebras and the Real and Rational Homotopy Type of Closed Manifolds

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 type of closed simply connected manifolds M . We prove that if M is a closed (r − 1)-connected manifold of dimension n ≤ l(r − 1) + 2 (where r ≥ 2, l ≥ 4), the real and rational homotopy type of M is defined uniquely by the cohomology algebra H*(M, F) and the isotopy modulo (l − 2) of the corresponding minimal unital cyclic C∞ -algebra enhancements of H*(M, F) for F = R, Q, respectively. Combining this with the Hodge homotopy introduced by Fiorenza-Kawai-Lê-Schwachhöfer, we provide a new proof of a theorem by Crowley-Nordström: a (r −1)-connected closed manifold M of dimension 4r − 1 with b_r (M) ≤ 3 is intrinsically formal if there exists a φ ∈ H^(2r−1) (M, R) such that the map H^r (M, R) → H^(3r−1) (M, R), x → φ ∪ x is an isomorphism. Furthermore, we provide a new proof and extension of Cavalcanti’s result, showing that a (r − 1)-connected closed manifold M of dimension 4r with b_r (M) ≤ 2 is intrinsically formal under similar conditions. This talk is based on arxiv.org/abs/2603.01219

mathematical physicsalgebraic topologydifferential geometryrepresentation theorystatistics theory

Audience: researchers in the topic

( slides | video )

Prague-Hradec Kralove seminar Cohomology in algebra, geometry, physics and statistics

Series comments: Virtual coffee starts on Zoom already 15 minutes before the seminar.