Differential Geometry Seminar: Scalable spaces

Seminar | October 14 | 3-4 p.m. | 939 Evans Hall | Note change in date

 Fedor Manin, UCSB

 Department of Mathematics

Given a Riemannian manifold $M$, harmonic forms induce a map $H^*(M;\mathbb R) \to \Omega ^*(M)$ which is in general not multiplicative. Manifolds for which it is are called geometrically formal and besides compact symmetric spaces few examples are known. A different picture emerges when we ask about the existence of some multiplicative map $H^*(M;\mathbb R) \to \Omega ^*(M)$. For example, such a map exists for $(\mathbb {CP} )^{\sharp 3}$ but not $(\mathbb {CP}^2)^{\sharp 4}$. Unlike geometric formality, this property is purely topological, in fact an invariant of rational homotopy type. It also has a number of geometric and topological consequences and equivalent formulations. This is joint work with Sasha Berdnikov.