Bundles in the category of Frölicher spaces and symplectic structure

Abstract

Bundles and morphisms between bundles are defined in the category of Fr¨olicher spaces (earlier known as the category of smooth spaces, see [2], [5], [9], [6] and [7]). We show that the sections of Fr¨olicher bundles are Fr¨olicher smooth maps and the fibers of Fr¨olicher bundles have a Fr¨olicher structure. We prove in detail that the tangent and cotangent bundles of a n-dimensional pseudomanifold are locally diffeomorphic to the even-dimensional Euclidian canonical F-space R2n. We define a bilinear form on a finite-dimensional pseudomanifold. We show that the symplectic structure on a cotangent bundle in the category of Fr¨olicher spaces exists and is (locally) obtained by the pullback of the canonical symplectic structure of R2n. We define the notion of symplectomorphism between two symplectic pseudomanifolds. We prove that two cotangent bundles of two diffeomorphic finite-dimensional pseudomanifolds are symplectomorphic in the category of Frölicher spaces.