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Homology of Normal Chains and Cohomology of Charges

About this Title

Th. De Pauw, R. M. Hardt and W. F. Pfeffer

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

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Table of Contents


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