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CHAPTER Ⅰ.TOPOLOGICAL INVARIANTS OF NON-HOMOTOPIC TYPE OF A FINITE POLYTOPE 1

1.The notion of complexes 1

2.Regular pairs of complexes and polytopes 9

3.Topological invariants of regular pairs of finite polytopes 12

4.Regular pairs associated to a finite polytope 21

5.Remarks 28

CHAPTER Ⅱ.THEORY OF P.A.SMITH ABOUT SPACES UNDER PERIODIC TRANSFORMATIONS WITH NO FIXED POINTS 35

1.Complexes with transformation groups 35

2.Complexes under periodic transformations 44

3.Smith homomorphisms and their properties 58

4.Spaces with transformation groups 71

5.Examples 79

CHAPTER Ⅲ.A GENERAL METHOD FOR THE STUDY OF IMBEDDING，IMMERSION AND ISOTOPY 92

1.Fundamental concepts 92

2.The Φp- and Ψp -classes of a finite polytope 101

3.Examples 112

4.Isotopy and isoposition 121

CHAPTER Ⅳ.CONDITIONS OF IMBEDDING AND IMMERSION IN TERMS OF COHOMOLOGY OPERATIONS 128

1.Smith theory of complexes under periodic transformations with invariant subcomplexes 128

2.Special homologies in product complexes 140

3.Smith operations 153

4.Conditions of imbedding and immersion in terms of smith operations 162

5.Relations between smith operations and steenrod powers 166

CHAPTER Ⅴ.THEORY OF OBSTRUCTIONS FOR THE IMBEDDING， IMMER-SION， AND ISOTOPY OF COMPLEXES IN A EUCLIDEAN SPACE 172

1.Linear realization of a complex in a euclidean space 172

2.Intersections and linkings in euclidean spaces 175

3.Obstruction to imbeddings of a complex in euclidean spaces 181

4.The realization of a cocycle in the imbedding class as an imbedding cocycle 186

5.The coincidence of imbedding classes ΦNK with the Φ2 -classes ΦN2 （K） of a finite simplicial complex K 190

6.Obstruction to immersion of a complex in a euclidean space 196

7.Obstruction to isotopy of imbeddings in a euclidean space 198

CHAPTER Ⅵ.SUFFICIENCY THEOREMS FOR THE IMBEDDING， IMMER-SION， AND ISOTOPY IN A EUCLIDEAN SPACE 208

1.Some elementary sufficiency theorems 208

2.Some fundamentals about C∞-maps 212

3.Some auxiliary geometric constructions 223

4.The main theorem for imbedding-necessary and sufficient conditions for Kn ？ R2n， n ＞ 2 233

5.The main theorem for immersion-necessary and sufficient conditions for Kn ？ R2n-1， n ＞ 3 239

6.The main theorem for isotopy-necessary and sufficient conditions for f， g： Kn ？ R 2n＋1， n＞ 1， to be isotopic 243

CHAPTER Ⅶ.IMBEDDING， IMMERSION， AND ISOTOPY OF MANIFOLDS IN A EUCLIDEAN SPACE 252

1.Periodic transformations in combinatorial manifolds 252

2.Sufficiency theorems for combinatorial manifolds 255

3.Imbedding of a combinatorial manifold 259

4.Immersion of a combinatorial manifold 266

5.An extension of the general theory in the case of differen-tial manifolds 274

Bibliographical Notes 283

Bibliography 288

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