Topological vector spaces problems in homology and homotopy theories

Abstract

Three data are interesting here: domains of integration, integrands and integration itself. There is a lack of symmetry between polyhedral chains as domains of integration and differential forms as integrands. The non-symmetric situation disappears after considering the topological spaces of the de Rham differential forms and forms with compact supports and their strong duals, i.e., currents with compact supports and currents, respectively. This idea goes back to Schwartz distributions and Schwartz distributions with compact supports, in other terminology, generalized functions and generalized functions with compact supports. Some problems are raised, e.g., whether every quasi-complete barreled nuclear space E, whose strongly dual E ′ is nuclear, is strongly hereditary reflexive. This concerns the above mentioned de Rham spaces. Problems on R - and Q -homotopy, proper R - and Q -homotopy and proper R - and Q -homotopy at infinity are also considered as well as the coalgebra structure on currents and currents with compact supports. The classical theorem concerning derivation of additive functions with respect to volumes in points is generalized to a theorem on derivation of continuous m-forms with compact supports ω m of an oriented n-dimensional C 1 -manifold M n with respect to its m-dimensional oriented submanifolds V m in compact regular oriented submanifolds L k of M n , 0 ⩽ k < m ⩽ n .