Given an open cover of a closed symplectic manifold, consider all smooth partitions of unity consisting of functions supported in the covering sets. The Poisson bracket invariant of the cover measures how much the functions from such a partition of unity can become close to being Poisson commuting. We introduce a new approach to this invariant, which enables us to prove the lower bound conjectured by L. Polterovich, in dimension 2.
All Science Journal Classification (ASJC) codes
- Partition of unity
- Poisson bracket invariant
- Poisson non-commutativity
- Symplectic surface