Bundles in the category of Frölicher spaces and symplectic structure

[摘要] Bundles and morphisms between bundles are defined in the category ofFr¨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¨olichersmooth 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-dimensionalpseudomanifold are locally diffeomorphic to the even-dimensional Euclidiancanonical F-space R2n. We define a bilinear form on a finite-dimensionalpseudomanifold. We show that the symplectic structure on a cotangent bundlein the category of Fr¨olicher spaces exists and is (locally) obtained by thepullback of the canonical symplectic structure of R2n. We define the notionof symplectomorphism between two symplectic pseudomanifolds. We provethat two cotangent bundles of two diffeomorphic finite-dimensional pseudomanifoldsare symplectomorphic in the category of Frölicher spaces.

[发布机构] University of the Witwatersrand