2017;
165 pp;
Softcover

MSC: Primary 57;

Print ISBN: 978-1-4704-3715-2

Product Code: MBK/109

List Price: $49.00

AMS Member Price: $39.20

MAA Member Price: $44.10

**Electronic ISBN: 978-1-4704-4305-4
Product Code: MBK/109.E**

List Price: $49.00

AMS Member Price: $39.20

MAA Member Price: $44.10

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#### Supplemental Materials

# Topology as Fluid Geometry: Two-Dimensional Spaces, Volume 2

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*James W. Cannon*

This is the second of a three volume collection devoted to the
geometry, topology, and curvature of 2-dimensional spaces. The
collection provides a guided tour through a wide range of topics by
one of the twentieth century's masters of geometric topology. The
books are accessible to college and graduate students and provide
perspective and insight to mathematicians at all levels who are
interested in geometry and topology.

The second volume deals with the topology of 2-dimensional
spaces. The attempts encountered in Volume 1 to understand length and
area in the plane lead to examples most easily described by the
methods of topology (fluid geometry): finite curves of infinite
length, 1-dimensional curves of positive area, space-filling curves
(Peano curves), 0-dimensional subsets of the plane through which no
straight path can pass (Cantor sets), etc. Volume 2 describes such
sets. All of the standard topological results about 2-dimensional
spaces are then proved, such as the Fundamental Theorem of Algebra
(two proofs), the No Retraction Theorem, the Brouwer Fixed Point
Theorem, the Jordan Curve Theorem, the Open Mapping Theorem, the
Riemann-Hurwitz Theorem, and the Classification Theorem for Compact
2-manifolds. Volume 2 also includes a number of theorems usually
assumed without proof since their proofs are not readily available,
for example, the Zippin Characterization Theorem for 2-dimensional
spaces that are locally Euclidean, the Schoenflies Theorem
characterizing the disk, the Triangulation Theorem for 2-manifolds,
and the R. L. Moore's Decomposition Theorem so useful in understanding
fractal sets.

#### Readership

Undergraduate and graduate students and researchers interested in topology.

#### Reviews & Endorsements

Many readers will be hooked by Cannon's aesthetics and proof exposition, where geometric intuition and topological arguments play leading roles...Cannon's books are worth every cent. I have in the past gifted Hilbert & Cohn-Voseen and Rademacher and Toeplitz to my students. Now I have Cannon's trio to add to my list of giftables.

-- Tushar Das, MAA Reviews

#### Table of Contents

# Table of Contents

## Topology as Fluid Geometry: Two-Dimensional Spaces, Volume 2

- Cover Cover11
- Title page iii4
- Contents v6
- Preface to the Three Volume Set ix10
- Preface to Volume 2 xiii14
- Chapter 1. The Fundamental Theorem of Algebra 116
- Chapter 2. The Brouwer Fixed Point Theorem 1126
- Chapter 3. Tools 2944
- Chapter 4. Lebesgue Covering Dimension 3752
- Chapter 5. Fat Curves and Peano Curves 4560
- Chapter 6. The Arc, the Simple Closed Curve, and the Cantor Set 5772
- Chapter 7. Algebraic Topology 7590
- Chapter 8. Characterization of the 2-Sphere 8196
- Chapter 9. 2-Manifolds 91106
- Chapter 10. Arcs in \St Are Tame 95110
- Chapter 11. R. L. Moore’s Decomposition Theorem 101116
- Chapter 12. The Open Mapping Theorem 109124
- Chapter 13. Triangulation of 2-Manifolds 113128
- Chapter 14. Structure and Classification of 2-Manifolds 117132
- Chapter 15. The Torus 129144
- Chapter 16. Orientation and Euler Characteristic 139154
- Chapter 17. The Riemann-Hurwitz Theorem 151166
- Bibliography 159174
- Back Cover Back Cover1181