Homology of normal chains and cohomology of charges
Author(s)
Bibliographic Information
Homology of normal chains and cohomology of charges
（Memoirs of the American Mathematical Society, no. 1172）
American Mathematical Society, c2017
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Note
"Volume 247, number 1172 (fifth of 7 numbers), May 2017"
Includes bibliographical references (p. 113-115)
Description and Table of Contents
Description
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.
Table of Contents
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
by "Nielsen BookData"
