Homology of normal chains and cohomology of charges - De Pauw, Th. (Thierry), 1971-; Hardt, R. (Robert), 1945-; Pfeffer, Washek F. (Memoirs of the American Mathematical Society ;, 2017)
We consider a category of pairs of compact metric spaces and Lipschitz maps where the pairs satisfy a linearly isoperimetric condition related to the solvability of the Plateau problem with partially free boundary. It includes properly all pairs of compact Lipschitz neighborhood retracts of a large class of Banach spaces. On this category we define homology and cohomology functors with real coefficients which satisfy the Eilenberg-Steenrod axioms, but reflect the metric properties of the underlying spaces. As an example we show that the zero-dimensional homology of a space in our category is trivial if and only if the space is path connected by arcs of finite length. The homology and cohomology of a pair are, respectively, locally convex and Banach spaces that are in duality. Ignoring the topological structures, the homology and cohomology extend to all pairs of compact metric spaces. For locally acyclic spaces, we establish a natural isomorphism between our cohomology and the Čech cohomology with real coefficients.