Homology of Normal Chains and Cohomology of Charges

ISBN-13: 9781470423353 / Angielski

Th. De Pauw;R. M. Hardt;W. F. Pfeffer

Opis

The authors 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 the authors 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 they 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, the authors establish a natural isomorphism between their cohomology and the Cech cohomology with real coefficients.

Spis treści

• Introduction
• Notation and preliminaries
• Rectifiable chains
• Lipschitz chains
• Flat norm and flat chains
• The lower semicontinuity of slicing mass
• Supports of flat chains
• Flat chains of finite mass
• Supports of flat chains of finite mass
• Measures defined by flat chains of finite mass
• Products
• Flat chains in compact metric spaces
• Localized topology
• Homology and cohomology
• $q$-bounded pairs
• Dimension zero
• Relation to the Cech cohomology
• Locally compact spaces
• References