Equivariant classification of $b^m$-symplectic surfaces and Nambu structures

Abstract

In this paper we extend the classification scheme in [S] for $b^m$-symplectic surfaces and, more generally, $b^m$-Nambu structures to the equivariant setting. When the compact group is the group of deck-transformations of an orientable covering, this yields the classification of these objects for non-orientable surfaces. The paper also includes recipes to construct $b^m$-symplectic structures on surfaces. Feasibility of such constructions depends on orientability and on the colorability of an associated graph. The desingularization technique in [GMW] is revisited for surfaces and the compatibility with this classification scheme is analyzed. We recast the strategy used in [Mt] to classify stable Nambu structures of top degree on orientable manifolds to classify $b^m$-Nambu structures (not necessarily oriented) using the language of $b^m$-cohomology. The paper ends up with an equivariant classification theorem of $b^m$-Nambu structures of top degree.