vm Preface Chapter 1 considers the simplest topological objects, graphs (one-dimen- sional complexes). First, we discuss questions which border on geometry, such as planarity, the Euler formula, and Steinitz' theorem. Then, we con- sider fundamental groups and coverings, whose basic properties are well seen in graphs. This chapter is concluded with a detailed discussion of the poly- nomial invariants of graphs there has been much interest in them recently, after the discovery of their relationship with knot invariants. Chapter 2 is concerned with another fairly simple topological object, Euclidean space with standard topology. Subsets of Euclidean space may have very complicated topological structure for this reason, only a few basic statements about the topology of Euclidean space and its subsets are included. One of the fundamental problems in topology is the classi- fication of continuous maps between topological spaces (on the spaces cer- tain constraints may be imposed the classification is up to some equiva- lence) . The simplest classifications of this kind are related to curves in the plane, i.e., maps of S1 to M?. First, we prove the Jordan theorem and the Whitney-Graustein classification theorem for smooth closed curves up to regular homotopy. Then, we prove the Brouwer fixed point theorem and Sperner's lemma by several different methods (in addition to the standard statement of Sperner's lemma, we give its refined version, which takes into account the orientations of simplices). We also prove the Kakutani fixed point theorem, which generalizes the theorem of Brouwer. The chapter is concluded by the Tietze theorem on extension of continuous maps, which is derived from Urysohn's lemma, and two theorems of Lebesgue, the open cover theorem, which is used in the rigorous proofs of many theorems from homotopy and homology theories, and the closed cover theorem, on which the definition of topological dimension is based. Chapter 3 begins with elements of general topology it gives the minimal necessary information constantly used in algebraic topology. We consider three properties (Hausdorffness, normality, and paracompactness) which substantially facilitate the study of topological spaces. Then, we consider two classes of topological spaces that are most important in algebraic topol- ogy (namely, simplicial complexes and CW-complexes), describe techniques for dealing with them (cellular and simplicial approximation), and prove that these spaces have the three properties mentioned above. We also introduce the notion of degree for maps of pseudomanifolds and apply it to prove the Borsuk-Ulam theorem, from which we derive many corollaries. The chapter is concluded with a description of some constructions of topological spaces, including joins, deleted joins, and symmetric products. We apply deleted joins to prove that certain n-dimensional simplicial complexes cannot be embedded in R2n.

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