Contents
§1 The Problem of realization or imbedding (iii)
§2 Analysis of some known methods (iii)
§3 Method of this book (ix)
§4 Structure of the book (xii)
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)
CHAPTER III A GENERAL METHOD FOR THE STUDY OF IMBEDDING, IMMERSION AND ISOTOPY (92)
§1 Fundamental concepts (92)
§2 The * and * -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 IMMERsmN 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 * with the * -classes * 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 IMMERsmN, AND IsoTOPY IN A EucLmEAN 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 condition for Kn*R2n-1 n>3 (239)
§6 The main theorem for isotopy-necessary and sufficient conditions for f,g: Kn*R2n+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 differential manifolds (274)
Bibliographical Notes (283)
Bibliography (288)