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Homology of Normal Chains and Cohomology of Charges

About this Title

Th. De Pauw, Université Denis Diderot, Institut de Mathématiques de Jussieu, Equipe de Géométrie et Dynamique, Bâtiment Sophie Germain, Case 7012, 75205 Paris Cedex 13, France, R. M. Hardt, Department of Mathematics, Rice University, Houston, Texas 77251 and W. F. Pfeffer, Department of Mathematics, University of California, Davis, California 95616 – and – University of Arizona, Tucson, Arizona 85721

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 247, Number 1172
ISBNs: 978-1-4704-2335-3 (print); 978-1-4704-3705-3 (online)
Published electronically: January 12, 2017
Keywords: Flat chains, normal chains, charges, homology, cohomology
MSC: Primary 49Q15, 55N35

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Table of Contents


  • Introduction
  • 1. Notation and preliminaries
  • 2. Rectifiable chains
  • 3. Lipschitz chains
  • 4. Flat norm and flat chains
  • 5. The lower semicontinuity of slicing mass
  • 6. Supports of flat chains
  • 7. Flat chains of finite mass
  • 8. Supports of flat chains of finite mass
  • 9. Measures defined by flat chains of finite mass
  • 10. Products
  • 11. Flat chains in compact metric spaces
  • 12. Localized topology
  • 13. Homology and cohomology
  • 14. $q$-bounded pairs
  • 15. Dimension zero
  • 16. Relation to the Čech cohomology
  • 17. Locally compact spaces


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.

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