2019/04/23 by Jiawei Zhou, Zhou, Jiawei
Mathematics · #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1904.10143
openalex publication_date 2019/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The rational homotopy type of a differential graded algebra (DGA) can be represented by a family of tensors on its cohomology, which constitute an A_∞-minimal model of this DGA. When only the cohomology is needed to determine the rational homotopy type, then the DGA is called formal. By a theorem of Miller, a compact k-connected manifold is formal if its dimension is not greater than 4k+2. We expand this theorem and a result of Crowley-Nordström to prove that if the dimension of a compact k-connected manifold N≤ (l+1)k+2, then its de Rham complex has an A_∞-minimal model with mp=0 for all p≥ l. Separately, for an odd-dimensional sphere bundle over a formal manifold, we prove that its de Rham complex has an A_∞-minimal model with only m2 and m3 non-trivial. In the special case of a circle bundle over a formal symplectic manifold satisfying the hard Lefschetz property, we give a necessary condition for formality which becomes sufficient when the base symplectic manifold is of dimension six or less.