eBook ISBN:  9781470437053 
Product Code:  MEMO/247/1172.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 
eBook ISBN:  9781470437053 
Product Code:  MEMO/247/1172.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 247; 2017; 115 ppMSC: Primary 49; 55
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 EilenbergSteenrod axioms, but reflect the metric properties of the underlying spaces.
As an example they show that the zerodimensional 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 Čech cohomology with real coefficients.

Table of Contents

Chapters

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


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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 EilenbergSteenrod axioms, but reflect the metric properties of the underlying spaces.
As an example they show that the zerodimensional 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 Čech cohomology with real coefficients.

Chapters

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