Topology.

*(English. Russian original)*Zbl 0839.55001
Topology I. Encycl. Math. Sci. 12, 1-310 (1996); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 12, 5-252 (1986).

This survey constitutes the introduction to a series of essays on topology, in which the development of its various subdisciplines is exposed in greater detail. For the Russian original edition see Zbl 0668.55001. The table of Contents: Introduction and Introduction to the English edition (both introductions signed by the author); Chapter 1: The simplest topological properties; Chapter 2. Topological spaces, fibrations, homotopies; Chapter 3: Simplicial complexes and CW-complexes, homology and cohomology, their relation to homotopy theory, obstructions; Chapter 4: Smooth Manifolds; Concluding remarks; Appendix: Recent developments in the topology of 3-manifolds and knots; Bibliography; Index.

In Chapter 1 the author defines topology as “the study of topological invariants of various kinds of mathematical objects, starting with rather general geometrical figures. From the topological point of view the name “geometrical figures” signifies: general polyhedra (polytopes) of various dimensions (complexes; or continuous or smooth “surfaces” of any dimension situated in some Euclidean space or regarded as existing independently (manifolds); or sometimes subsets of a more general nature of a Euclidean space or a manifold, or even of an infinite-dimensional space of functions”. The author gives some familiar and elementary results: Euler’s theorem, Gauss-Bonnet formula but also some special examples as nontrivially linked curves etc.

Chapter 2 contains: observations from general topology, homotopies and homotopy type, covering homotopies and fibrations.

In Chapter 3 the author studies simplicial complexes, the homology and cohomology groups together with the Poincaré duality, relative homology, the exact sequence of a pair, the axioms for a homology theory and CW-complexes (Hopf’s theorem and Hurewicz’s theorem), simplicial complexes and other homology theories, singular homology, coverings and sheaves, the exact sequence of sheaves and cohomology, homology theory of non-simply connected spaces, complexes of modules, Reidemeister torsion, simple homotopy type, simplicial and cell bundles with a structure group, obstructions, universal objects: universal fiber bundles and the universal property of Eilenberg-MacLane complexes, cohomology operations, the Steenrod algebra, the Adams spectral sequence, the “classical” apparatus of homotopy theory, the Leray spectral sequence, the homology theory of fiber bundles, the Cartan-Serre method, the Postnikov tower, the Adams spectral sequence, 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, 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 contains: basic concepts on smooth manifolds, smooth fiber bundles, connexions and characteristic classes, smooth manifolds and homotopy theory, framed manifolds, bordisms, Thom spaces, the Hirzebruch formulae, estimates of the orders of homotopy groups of spheres, Milnor’s example, the integral properties of cobordism, classification problems in the theory of smooth manifolds, the theory of immersions, manifolds with the homotopy types of a sphere, relationships between smooth and PL-manifolds, integral Pontryagin classes, the role of the fundamental group in topology, manifolds of low dimension \((n = 2,3)\), knots, the boundary of an open manifold, the topological invariance of the rational Pontryagin classes, the classification theory of nonsimply connected manifolds of dimension \(\geq 5\), higher signatures, Hermitian \(K\)-theory, geometric topology: the construction of nonsmooth homeomorphisms, Milnor’s example, the annulus conjecture, topological and PL-structures.

In the Concluding Remarks the author emphasizes the achievements of the book “In this survey the ideas and methods of topology far from all topics have been considered” besides some omissions (in the author’s opinion) “\(\dots\) it has not proved possible to discuss the substantial applications of topology that have been made over recent decades to real physical problems, and have transformed the apparatus of modern mathematical physics. We hope that this lack will be made good in other essays of the series”.

In the Appendix the author discusses recent developments in topology: knots (the classical and modern approaches to the Alexander polynomial and Jones-type polynomials, Vassiliev invariants, new topological invariants for 3-manifolds, topological quantum field theories).

The Bibliography contains: I. Popular books and articles on geometry, topology and their applications (books of the 1930’s, modern popular geometric-topological books, recent popular articles written by or together with physicists), II. Text-books on combinatorial and algebraic topology. III Books on elementary and PL-topology of manifolds, complex manifold theory, fibration geometry, Lie groups (elementary differential topology, PL-manifold topology, forms, sheaves, complex and algebraic manifolds, foundations of differential geometry and topology: fibre bundles, Lie groups). IV. Surveys and textbooks on particular aspects of topology and its applications (variational calculus “in the large”, knot theory, theory of finite and compact transformation groups, homotopy theory, cohomology operations, spectral sequence, theory of characteristic classes and cobordisms, \(K\)-theory and the index of elliptic operators, algebraic \(K\)-theory, multiply connected manifolds, categories and functors, homological algebra, general questions of homotopy theory, four-dimensional manifolds, manifolds of few dimensions, classification problems of higher-dimensional topology of manifolds, singularities of smooth functions and maps, foliations, cohomology of Lie algebras of vector fields). V. Selected research works and monographs to the same subjects as in IV.

The book gives an overview of combinatorial, algebraic, differential, homotopic, and geometric topology, beginning with the elements and proceeding right up to the present frontiers of research.

For the entire collection see [Zbl 0830.00014].

In Chapter 1 the author defines topology as “the study of topological invariants of various kinds of mathematical objects, starting with rather general geometrical figures. From the topological point of view the name “geometrical figures” signifies: general polyhedra (polytopes) of various dimensions (complexes; or continuous or smooth “surfaces” of any dimension situated in some Euclidean space or regarded as existing independently (manifolds); or sometimes subsets of a more general nature of a Euclidean space or a manifold, or even of an infinite-dimensional space of functions”. The author gives some familiar and elementary results: Euler’s theorem, Gauss-Bonnet formula but also some special examples as nontrivially linked curves etc.

Chapter 2 contains: observations from general topology, homotopies and homotopy type, covering homotopies and fibrations.

In Chapter 3 the author studies simplicial complexes, the homology and cohomology groups together with the Poincaré duality, relative homology, the exact sequence of a pair, the axioms for a homology theory and CW-complexes (Hopf’s theorem and Hurewicz’s theorem), simplicial complexes and other homology theories, singular homology, coverings and sheaves, the exact sequence of sheaves and cohomology, homology theory of non-simply connected spaces, complexes of modules, Reidemeister torsion, simple homotopy type, simplicial and cell bundles with a structure group, obstructions, universal objects: universal fiber bundles and the universal property of Eilenberg-MacLane complexes, cohomology operations, the Steenrod algebra, the Adams spectral sequence, the “classical” apparatus of homotopy theory, the Leray spectral sequence, the homology theory of fiber bundles, the Cartan-Serre method, the Postnikov tower, the Adams spectral sequence, 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, 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 contains: basic concepts on smooth manifolds, smooth fiber bundles, connexions and characteristic classes, smooth manifolds and homotopy theory, framed manifolds, bordisms, Thom spaces, the Hirzebruch formulae, estimates of the orders of homotopy groups of spheres, Milnor’s example, the integral properties of cobordism, classification problems in the theory of smooth manifolds, the theory of immersions, manifolds with the homotopy types of a sphere, relationships between smooth and PL-manifolds, integral Pontryagin classes, the role of the fundamental group in topology, manifolds of low dimension \((n = 2,3)\), knots, the boundary of an open manifold, the topological invariance of the rational Pontryagin classes, the classification theory of nonsimply connected manifolds of dimension \(\geq 5\), higher signatures, Hermitian \(K\)-theory, geometric topology: the construction of nonsmooth homeomorphisms, Milnor’s example, the annulus conjecture, topological and PL-structures.

In the Concluding Remarks the author emphasizes the achievements of the book “In this survey the ideas and methods of topology far from all topics have been considered” besides some omissions (in the author’s opinion) “\(\dots\) it has not proved possible to discuss the substantial applications of topology that have been made over recent decades to real physical problems, and have transformed the apparatus of modern mathematical physics. We hope that this lack will be made good in other essays of the series”.

In the Appendix the author discusses recent developments in topology: knots (the classical and modern approaches to the Alexander polynomial and Jones-type polynomials, Vassiliev invariants, new topological invariants for 3-manifolds, topological quantum field theories).

The Bibliography contains: I. Popular books and articles on geometry, topology and their applications (books of the 1930’s, modern popular geometric-topological books, recent popular articles written by or together with physicists), II. Text-books on combinatorial and algebraic topology. III Books on elementary and PL-topology of manifolds, complex manifold theory, fibration geometry, Lie groups (elementary differential topology, PL-manifold topology, forms, sheaves, complex and algebraic manifolds, foundations of differential geometry and topology: fibre bundles, Lie groups). IV. Surveys and textbooks on particular aspects of topology and its applications (variational calculus “in the large”, knot theory, theory of finite and compact transformation groups, homotopy theory, cohomology operations, spectral sequence, theory of characteristic classes and cobordisms, \(K\)-theory and the index of elliptic operators, algebraic \(K\)-theory, multiply connected manifolds, categories and functors, homological algebra, general questions of homotopy theory, four-dimensional manifolds, manifolds of few dimensions, classification problems of higher-dimensional topology of manifolds, singularities of smooth functions and maps, foliations, cohomology of Lie algebras of vector fields). V. Selected research works and monographs to the same subjects as in IV.

The book gives an overview of combinatorial, algebraic, differential, homotopic, and geometric topology, beginning with the elements and proceeding right up to the present frontiers of research.

For the entire collection see [Zbl 0830.00014].

Reviewer: I.Pop (Iaşi)

##### MSC:

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

57R19 | Algebraic topology on manifolds and differential topology |

57R20 | Characteristic classes and numbers in differential topology |

57R75 | \(\mathrm{O}\)- and \(\mathrm{SO}\)-cobordism |

55N10 | Singular homology and cohomology theory |

55R05 | Fiber spaces in algebraic topology |

55S35 | Obstruction theory in algebraic topology |