How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

   1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

   1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

   2. Complete and sign the license agreement.

   3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2213  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046
Email: cust-serv@ams.org

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

Powered by MathJax
  Remote Access

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)
DOI: https://doi.org/10.1090/memo/1172
Published electronically: January 12, 2017
Keywords:Flat chains, normal chains, charges, homology, cohomology

View full volume PDF

View other years and numbers:

Table of Contents


Chapters

  • 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

Abstract


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.

References [Enhancements On Off] (What's this?)