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Cohomology in algebra, geometry, physicsand statistics
Almost formality of manifolds of low dimension
Speaker’s name:
Lorenz Schwachhoefer
Speaker’s affiliation:
TU Dortmund
Place:
in IM building, ground floor
Date:
Wednesday, 5. June 2019 - 11:30 to 12:30
Abstract:
This is based on work with Domenico Fiorenza, Kotaro Kawai and Hong Van Le.
A classical result by Miller states that any $(k-1)$-connected closed oriented manifold of dimension $\leq 4k-2$ is formal. More recently, Crowley and Nordstr\”om defined the Bianchi-Massey tensor and showed that its vanishing is the only obstruction to the formality of a $(k-1)$-connected closed oriented manifold of dimension $\leq 5k-3$. Moreover, recently Chan-Karigiannis-Tsang showed that closed $G_2$-manifolds are almost formal, meaning that their deRham algebra is equivalent to a DGA whose differential vanishes in all but one dimension.
We consider this question from a more general point of view and define Poincar\’e-Differential Graded Commutative Algebras (DGCAs) of Hodge type, a class which includes the deRham algebra of closed oriented manifolds. We show that in the simply connected case, each such algebra is quasi-equivalent to a finite dimensional Poincar\’e-DGCA, and regarding this as an $A_\infty$-algebra, we give a cohomological interpretation of the (vanishing of the) Bianchi-Massey tensor. We also show that any $(k-1)$-connected closed oriented manifold of dimension $\leq 4k-1$ is almost formal in the sense of Chan-Karigiannis-Tsang.
A classical result by Miller states that any $(k-1)$-connected closed oriented manifold of dimension $\leq 4k-2$ is formal. More recently, Crowley and Nordstr\”om defined the Bianchi-Massey tensor and showed that its vanishing is the only obstruction to the formality of a $(k-1)$-connected closed oriented manifold of dimension $\leq 5k-3$. Moreover, recently Chan-Karigiannis-Tsang showed that closed $G_2$-manifolds are almost formal, meaning that their deRham algebra is equivalent to a DGA whose differential vanishes in all but one dimension.
We consider this question from a more general point of view and define Poincar\’e-Differential Graded Commutative Algebras (DGCAs) of Hodge type, a class which includes the deRham algebra of closed oriented manifolds. We show that in the simply connected case, each such algebra is quasi-equivalent to a finite dimensional Poincar\’e-DGCA, and regarding this as an $A_\infty$-algebra, we give a cohomological interpretation of the (vanishing of the) Bianchi-Massey tensor. We also show that any $(k-1)$-connected closed oriented manifold of dimension $\leq 4k-1$ is almost formal in the sense of Chan-Karigiannis-Tsang.