Introduction
Introduction to the English Translation
Chapter 1. The Simplest Topological Properties
Chapter 2. topological Spaces. Fibrations. Homotopies
1. Observations from general topology. Terminology
2. Homotopies. Homotopy type
3. Covering homotopies. Fibrations
4. Homotopy groups and fibrations. Exact sequences. Examples
Chapter 3. Simplicial Complexes and CW-complexes. Homology and Cohomology. Their Relation to Homotopy Theory. Obstructions
1. Simplicial complexes
2. The homology and cohomology groups.Poincare duality
3. Relative homology. The exact sequence of a pair. Axioms for homology theory. CW-complexes
4. Simplicial complexes and other homology theories. Singular homulogy. Coverings and sheaves. The exact sequence of sheaves and cohomology
5. Homology theory of non-simply-connected spaces. Complexes of modules. Reidemeister torsion. Simples homotopy type
6. Simplicial and cell bundles with a structure group. Obstructions. Universal lbjects： universal fiber bundles and the universal property of Eilenberg-MacLane complexes. Cohomology operations. The Steenrod algebra. The Adams spectral sequence
7. The classical apparatus of homotopy theory. The Laray spectral sequence. The homology theory of fiber bundles. The Cartan-Serre method. The Postnikov tower. The Adams spectral sequence
8. Definition and properties of K-theory. The Atiyah-Hirzebruch spectral sequence. Adams operations. Analogues of the Thom isomorphism and the Riemann-Roch theorem. Elliptic operators and K-theory. Transformation groups. Four-dimensional manifolds
9. Bordism and cobordism theory as generalized homology and cohomology. Cohomology operations in cobordism. The Adams-Novikov spectral sequence. Formal groups. Actions of cyclic groups and the circle on manifolds
Chapter 4. Smooth Manifolds
1. Basic concepts. Smooth fiber bundles. Connexions. Characteristic classes
2. The homology theory of smooth manifolds. Complex manifolds. The classical global calculus of variations. H-spaces. Multi-valued functions and functionals
3. Smooth manifolds and homotopy theory. Framed manifolds. Bordisms. Thom spaces. The Hirzebruch formulae. Estimates of the orders of homotopy groups of spheres. Milnors example. The integral properties of cobordisms
4. Classification problems in the theory of smooth manifolds. The theory of immersions. Manifolds with the homotopy type of a sphere. Relationships between smooth and PL-manifolds. Integral Pontryagin classes
5. The role of the fundamental group in topology. Manifolds of low dimension (n=2，3). Knots. The boundary of an open manifold. The classification invariance of the rational Pontryagin classes. The classification theory of non-simply-connected manifolds of dimension>5. Higher signatures. Hermitian K-theory. Geometric topology： the construction of non-smooth homeomorphisms. Milnors example. The annulus conjecture. Topological and PL-structures
Concluding Remarks
Appendix. Recent Developments in the Topology of 3-manifolds and Knots
1. Introduction： Recent developments in Topology
2. Knots： the classical and modern approaches to the Alexander polynomial. Jones-type polynomials
3. Vassiliev Invariants
4. New topological invariants for 3-manifolds. Topological Quantum Field Theories
Bibliography
Index