Topological methods in Euclidean spaces 欧氏间中的拓扑方法

The author of this wide-ranging introduction to point-set, algebraic, and differential topology has avoided the tiresome technicalities inherent in axiomatic treatments by limiting his study to Euclidean subspaces. This approach has also enabled him to develop extensively a number of topics central to topology itself as well as its applications in other areas of mathematics and science.
　　In addition to the basic point-set topology of Euclidean spaces, the book covers elementary combinatorial techniques, including barycentric subdivisions, Sperner's Lemma, and the Brouwer Fixed Point Theorem; homotopy theory and the fundamental group, which examines--among other topics--maps of spheres and fundamental groups of the spheres; simplicial homology theory and homology groups of topological polyhedra, the Hopf Trace Theorem, and the Lefschetz Fixed Point Theorem; and differential techniques, including the Stone-Weierstrass Theorem, Morse functions, and smooth tan- gent vector fields.
　　A new section of solutions to selected problems has been appended to this affordable edition, sure to be welcomed by advanced under-graduate and graduate students of mathematics.

Preface
Chapter　1　Point-set topology of Euclidean spaces
　1　Introduction
　2　Preliminaries
　3　Open sets, closed sets, and continuity
　4　Compact spaces
　5　Connectivity properties
　6　Real-valued continuous functions
　7　Retracts
　8　Topological dimension
　Supplementary exercises
Chapter 2　Elementary combinatorial techniques
　1　Introduction
　2　Hyperplanes in Rn
　3　Simplexes and complexes
　4　Sample triangulations
　5　Simplicial maps
　6　Barycentric subdivision
　7　The Simplicial Approximation Theorem
　8　Sperner's Lemma
　9　The Brouwer Fixed Point Theorem
　10　Topological dimension of compact subsets of R
　Supplementary exercises
Chapter　3　Homotopy theory and the fundamental group
　1　Introduction
　2　The homotopy relation, nullhomotopic maps, and contractible spaces
　3　Maps of spheres
　4　The fundamental group
　5　Fundamental groups of the spheres
　Supplementary exercises
Chapter　4　Simplicial homology theory
　1　Introduction
　2　Oriented complexes and chains
　3　Boundary operators
　4　Cycles, boundaries, and homology groups
　5　Elementary examples
　6　Cone complexes, augmented complexes, and the homology groups Hp (K(Sn+l)
　7　Incidence numbers and the homology groups Hp(Kn(Sn+1)
　8　Elementary homological algebra
　9　The homology complex of a geometric complex
　10　Acyclic carrier functions
　11　Invariance of homology groups under barycentric subdivision
　12　Homomorphisms induced by continuous maps
　13　Homology groups of topological polyhedra
　14　The Hopf Trace Theorem
　15　The Lefschetz Fixed Point Theorem
　　Supplementary exercises
Chapter　5　Differential techniques
　1　Introduction
　2　Smooth maps
　3　The Stone-Weierstrass Theorem
　4　Derivatives as linear transformations
　5　Differentiable manifolds
　6　Tangent spaces and derivatives
　7　Regular and critical values of smooth maps
　8　Measure zero and Sard's Theorem
　9　Morse functions
　10　Manifolds with boundary
　11　One-dimensional manifolds
　12　Topological characterization of Sk
　13　Smooth tangent vector fields
　　Supplementary exercises
Solutions to Selected Exercises
Guide to further study
Bibliography
List of symbols and notation
Index