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The rational homotopy type of (n-1)-connected manifolds of dimension up to 5n-3

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arxiv 1505.04184 v3 pith:544X55ZV submitted 2015-05-15 math.AT math.GT

The rational homotopy type of (n-1)-connected manifolds of dimension up to 5n-3

classification math.AT math.GT
keywords tensorbianchi-masseyconnectedmanifoldsdimensionformalhomotopyrational
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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We define the Bianchi-Massey tensor of a topological space X to be a linear map from a subquotient of the fourth tensor power of H*(X). We then prove that if M is a closed (n-1)-connected manifold of dimension at most 5n-3 (and n > 1) then its rational homotopy type is determined by its cohomology algebra and Bianchi-Massey tensor, and that M is formal if and only if the Bianchi-Massey tensor vanishes. We use the Bianchi-Massey tensor to show that there are many (n-1)-connected (4n-1)-manifolds that are not formal but have no non-zero Massey products, and to present a classification of simply-connected 7-manifolds up to finite ambiguity.

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

    math.AT 2026-03 conditional novelty 6.0

    For (r−1)-connected closed manifolds of dimension n ≤ l(r−1)+2, rational homotopy type is determined by the cohomology ring and the minimal cyclic C∞-algebra enhancement up to isotopy modulo l−2.