6 feb 2020 -- 14:30
Aula Tricerri, DiMaI, Firenze
Seminario svolto nell'ambito del progetto PRIN 2017 "Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics", codice 2017JZ2SW5.
Abstract.
I'll 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. Using the small algebra and the small quotient algebra of a Poincaré DGCA of Hodge type, we'll investigate equivalence classes of (r-1)-connected Poincaré DGCAs of Hodge type, for r>1. In particular, we'll see that a (r-1)-connected Poincaré DGCA of Hodge $A^*$ type of dimension $n\leq 5r−3$ is $A_\infty$-quasi-isomorphic to an $A_3$-algebra and will show that the only obstruction to the formality of the algebra is a distinguished Harrison cohomology class. Moreover, this cohomology class and the DGCA isomorphism class of $H^*(A^*)$ determine the $A_\infty$-quasi-isomorphism class of $A^*$. This can be seen as a Harrison cohomology version of the Crowley-Nordstrom results on the rational homotopy type of (r-1)-connected (closed manifolds of dimension up to 5r-3. Based on joint work with Kotaro Kawai, Hong Van Le, and Lorenz Schwachhofer (arXiv:1902.08406).