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Introduction 1

CHAPTER Ⅰ ELEMENTARY COMBINATORIAL THEORY OF COMPLEXES 7

1．The simplex,the cell,the sphere and the complex 7

2．Orientation．Chains．Cycles 14

3．Arithmetical digression on matrices and moduli 23

4．The Betti and torsion numbers of a complex and their calculation 34

5．Circuits and their orientation 44

6．Orientation of Sn 54

7．Convex complexes and their subdivision．Invariance of the homology characters under subdivision 58

CHAPTER Ⅱ TOPOLOGICAL INVARIANCE OF THE HOMOLOGY CHARACTERS 72

1．Topological homology characters 72

2．Deformations．The Poincaré Group 76

3．The fundamental deformation theorem for complexes 84

4．Invariance of the homology characters 87

5．Invariance under partial homeomorphism 90

6．Invariance of dimensionality,regionality and the simple circuit 97

7．Extension of boundary relations 102

8．Combinatorial complexes 105

CHAPTER Ⅲ MANIFOLDS AND THEIR DUALITY THEOREMS 115

1．The structure of the stars of a normal complex 116

2．Definition of manifolds and their general properties 119

3．Duality relations for the homology characters 135

4．Invariance of manifolds 155

CHAPTER Ⅳ INTERSECTIONS OF CHAINS ON A MANIFOLD 161

1．Polyhedral intersections 162

2．Special combinatorial properties of the Kronecker index 174

3．Intersections of arbitrary chains and their combinatorial invariance 182

4．Topological invariance of the intersection elements 198

5．Looping coefficients 205

6．A new definition of intersections of chains 210

7．Miscellaneous questions 216

CHAPTER Ⅴ PRODUCT COMPLEXES 220

1．Generalities on product sets and spaces 220

2．Product complexes 222

3．Products of manifolds 234

CHAPTER Ⅵ TRANSFORMATIONS OF MANIFOLDS,THEIR COINCIDENCES AND FIXED POINTS 244

1．Position of the problem 245

2．Representative cycles of the transformations．The signed coincidences or fixed points 247

3．Approximation to transformations of complexes 255

4．Transformations of the cycles 258

5．The numbers of signed coincidences and fixed points 268

6．Special properties of single-valued transformations 278

7．Extension to arbitrary complexes 281

CHAPTER Ⅶ INFINITE COMPLEXES AND THEIR APPLICATIONS 291

1．General properties of infinite complexes 292

2．Ideal elements 295

3．Infinite manifolds and their duality theory 312

4．Compact metric spaces and their homology characters 323

5．Closed subsets of Sr 336

6．Transformations of compact metric spaces 343

CHAPTER Ⅷ APPLICATIONS TO ANALYTICAL AND ALGEBRAIC VARIETIES 361

1．Analytical varieties 362

2．Intersections of analytical varieties 369

3．Complex varieties 377

4．Applications to algebraic geometry 385

Bibliography 393

Addenda 408

Index 410