《Topology Second Edition》

Introduction1

CHAPTER ⅠELEMENTARY COMBINATORIAL THEORY OF COMPLEXES7

1．The simplex,the cell,the sphere and the complex7

2．Orientation．Chains．Cycles14

3．Arithmetical digression on matrices and moduli23

4．The Betti and torsion numbers of a complex and their calculation34

5．Circuits and their orientation44

6．Orientation of Sn54

7．Convex complexes and their subdivision．Invariance of the homology characters under subdivision58

CHAPTER ⅡTOPOLOGICAL INVARIANCE OF THE HOMOLOGY CHARACTERS72

1．Topological homology characters72

2．Deformations．The Poincaré Group76

3．The fundamental deformation theorem for complexes84

4．Invariance of the homology characters87

5．Invariance under partial homeomorphism90

6．Invariance of dimensionality,regionality and the simple circuit97

7．Extension of boundary relations102

8．Combinatorial complexes105

CHAPTER ⅢMANIFOLDS AND THEIR DUALITY THEOREMS115

1．The structure of the stars of a normal complex116

2．Definition of manifolds and their general properties119

3．Duality relations for the homology characters135

4．Invariance of manifolds155

CHAPTER ⅣINTERSECTIONS OF CHAINS ON A MANIFOLD161

1．Polyhedral intersections162

2．Special combinatorial properties of the Kronecker index174

3．Intersections of arbitrary chains and their combinatorial invariance182

4．Topological invariance of the intersection elements198

5．Looping coefficients205

6．A new definition of intersections of chains210

7．Miscellaneous questions216

CHAPTER ⅤPRODUCT COMPLEXES220

1．Generalities on product sets and spaces220

2．Product complexes222

3．Products of manifolds234

CHAPTER ⅥTRANSFORMATIONS OF MANIFOLDS,THEIR COINCIDENCES AND FIXED POINTS244

1．Position of the problem245

2．Representative cycles of the transformations．The signed coincidences or fixed points247

3．Approximation to transformations of complexes255

4．Transformations of the cycles258

5．The numbers of signed coincidences and fixed points268

6．Special properties of single-valued transformations278

7．Extension to arbitrary complexes281

CHAPTER ⅦINFINITE COMPLEXES AND THEIR APPLICATIONS291

1．General properties of infinite complexes292

2．Ideal elements295

3．Infinite manifolds and their duality theory312

4．Compact metric spaces and their homology characters323

5．Closed subsets of Sr336

6．Transformations of compact metric spaces343

CHAPTER ⅧAPPLICATIONS TO ANALYTICAL AND ALGEBRAIC VARIETIES361

1．Analytical varieties362

2．Intersections of analytical varieties369

3．Complex varieties377

4．Applications to algebraic geometry385

Bibliography393

Addenda408

Index410