The rational homotopy type of (n-1)-connected manifolds of dimension up to 5n-3
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2015
- DOI:
- 10.48550/arXiv.1505.04184
- arXiv:
- arXiv:1505.04184
- Bibcode:
- 2015arXiv150504184C
- Keywords:
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- Mathematics - Algebraic Topology;
- Mathematics - Geometric Topology;
- 55P62;
- 57N65
- E-Print:
- 31 pages. v3: Corrected statement of Theorem 1.5. To appear in the Journal of Topology