Cannon J.W. Two-Dimensional Spaces, Volume 2: Topology as Fluid Geometry

Providence, Rhode Island: American Mathematical Society, 2017. — 181 p. — ISBN10: 1470437155.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.
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.
Two-Dimensional Spaces, Volumes 1, 2, and 3
Volume 1
Lengths—The Pythagorean theorem
Consequences of the Pythagorean theorem
Areas
Areas by slicing and scaling
Areas by cut and paste
Areas by counting
Unsolvable problems in Euclidean geometry
Does every set have a size?Volume 2
The fundamental theorem of algebra
The Brouwer fixed point theorem
Tools
Lebesgue covering dimension
Fat curves and Peano curves
The arc, the simple closed curve, and the Cantor set
Algebraic topology
Characterization of the 2-sphere
2-manifolds
Arcs in $\mathbb{S}^2$ are tame
R. L. Moore's decomposition theorem
The open mapping theorem
Triangulation of 2-manifolds
Structure and classification of 2-manifolds
The torus
Orientation and Euler characteristic
The Riemann-Hurwitz theoremVolume 3
A graphical introduction to hyperbolic geometry
Hyperbolic geometry
Gravity as curvature
Curvature by polyhedral approximation
Curvature as a length derivative
Theorema egregium
Curvature appendix