Topology, K. Janich

For a number of years, beginning with the early 70's, the authors have been delivering lectures on the fundamentals of geometry and topology in the Faculty of Mechanics and Mathematics of Moscow State University. This text-book is the result of this work. We shall recall that for a long period of time the basic elements of modern geometry and topology were not included, even by departments and faculties of mathematics, as compulsory subjects in a university-level mathematical education. The standard courses in classical differential geometry have gradually become outdated, and there has been, hitherto, no unanimous standpoint as to which parts of modern geometry should be viewed as abolutely essential to a modern mathematical education. In view of the necessity of using a large number of geometric concepts and methods, a modernized course in geometry was begun in 1971 in the Mechanics division of the Faculty of Mechanics and Mathematics of Moscow State University. In addition to the traditional geometry of curves and surfaces, the course included the fundamental priniciples of tensor analysis, Riemannian geometry and topology. Some time later this course was also introduced in the division of mathematics. On the basis of these lecture courses, the following text-books appeared:

This book provides an accessible yet rigorous overview of current research in topology, making it an invaluable resource for mathematicians, physicists, and researchers in related fields.

Since Additive Manufacturing (AM) techniques allow the manufacture of complex-shaped structures the combination of lightweight construction, topology optimization, and AM is of significant interest. Besides the established continuum topology optimization methods, less attention is paid to algorithm-driven optimization based on linear optimization, which can also be used for topology optimization of truss-like structures.

General topology is the domain of mathematics devoted to the investigation of the concepts of continuity and passage to a limit at their natural level of generality. The most basic concepts of general topology, that of a topological space and a continuous map, were introduced by Hausdorff in 1914. One ofthe central problems oftopology is the determination and investigation of topological invariants; that is, properties of spaces which are preserved under homeomorphisms.

This textbook is intended for a course in algebraic topology at the beginning graduate level. The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory. These topics are developed systematically, avoiding all unnecessary definitions, terminology, and technical machinery. The text consists of material from the first five chapters of the author's earlier book, Algebraic Topology; an Introduction (GTM 56) together with almost all of his book, Singular Homology Theory (GTM 70). The material from the two earlier books has been substantially revised, corrected, and brought up to date.