In this paper we introduce the notion of Poincaré DGCAs of Hodge type, which is a subclass of Poincaré DGCAs encompassing the de Rham algebras of closed orientable manifolds. Then we introduce the notion of the small algebra and the small quotient algebra of a Poincaré DGCA of Hodge type. Using these concepts, we investigate the equivalence class of (r-1)-connected (r > 1) Poincaré DGCAs of Hodge type. In particular, we show that an (r-1)-connected Poincar ́e DGCA of Hodge type A* of dimension n <= 5r - 3 is A-infinity-quasi-isomorphic to an A_3-algebra and prove that the only obstruction to the formality of A* is a distinguished Harrison cohomology class [μ3] in Harr^3-1(H*(A*), H*(A*)). Moreover, the cohomology class [μ3] and the DGCA isomorphism class of H*(A*) determine the A-infinity-quasi-isomorphism class of A*. This can be seen as a Harrison cohomology version of the Crowley- Nordstrom results on rational homotopy type of (r - 1)-connected (r > 1) closed manifolds of dimension up to 5r -3. We also derive the almost formality of closed G2 -manifolds, which have been discovered recently by Chan-Karigiannis- Tsang, from our results and the Cheeger-Gromoll splitting theorem.
Almost formality of manifolds of low dimension / Fiorenza, D.; Kawai, K.; Van Le, H.; Schwachhofer, L.. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 2036-2145. - 22:1(2021), pp. 79-107. [10.2422/2036-2145.201905_002]
Almost formality of manifolds of low dimension
Fiorenza D.;
2021
Abstract
In this paper we introduce the notion of Poincaré DGCAs of Hodge type, which is a subclass of Poincaré DGCAs encompassing the de Rham algebras of closed orientable manifolds. Then we introduce the notion of the small algebra and the small quotient algebra of a Poincaré DGCA of Hodge type. Using these concepts, we investigate the equivalence class of (r-1)-connected (r > 1) Poincaré DGCAs of Hodge type. In particular, we show that an (r-1)-connected Poincar ́e DGCA of Hodge type A* of dimension n <= 5r - 3 is A-infinity-quasi-isomorphic to an A_3-algebra and prove that the only obstruction to the formality of A* is a distinguished Harrison cohomology class [μ3] in Harr^3-1(H*(A*), H*(A*)). Moreover, the cohomology class [μ3] and the DGCA isomorphism class of H*(A*) determine the A-infinity-quasi-isomorphism class of A*. This can be seen as a Harrison cohomology version of the Crowley- Nordstrom results on rational homotopy type of (r - 1)-connected (r > 1) closed manifolds of dimension up to 5r -3. We also derive the almost formality of closed G2 -manifolds, which have been discovered recently by Chan-Karigiannis- Tsang, from our results and the Cheeger-Gromoll splitting theorem.File | Dimensione | Formato | |
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