Preface IX Chapter 4 covers very diverse topics, such as two-dimensional surfaces, coverings, local homeomorphisms, graphs on surfaces (including genera of graphs and graph coloring), bundles, and homotopy groups. Chapter 5 turns to differential topology. We consider smooth manifolds and the application of smooth maps to topology. First, we introduce some basic tools (namely, smooth partitions of unity and Sard's theorem) and con- sider an example, the Grassmann manifolds, which plays an important role everywhere in topology. Then, we discuss notions related to tangent spaces, namely, vector fields and differential forms. After this, we prove existence theorems for embeddings and immersions (including closed embeddings of noncompact manifolds), which play an important role in the study of smooth manifolds. Moreover, we prove that a closed nonorientable n-manifold can- not be embedded in R n + 1 and determine what two-dimensional surfaces can be embedded in MP3. Further, we introduce a homotopy invariant, the degree of a smooth map, and apply it to define the index of a singular point of a vector field. We prove the Hopf theorem, which gives a homotopy classification of maps Mn — Sn. We also describe a construction of Pon- tryagin which interprets 7rn+/c(Sfn) as the set of classes of cobordant framed fc-manifolds in JRn+/\ We conclude this chapter with Morse theory, which relates the topological structure of a manifold to local properties of singular points of a nondegenerate function on this manifold. We give explicit exam- ples of Morse functions on some manifolds, including Grassmann manifolds. Chapter 6 is devoted to explicit calculations of fundamental groups for some spaces and to applications of fundamental groups. First, we prove a theorem about generators and relations determining the fundamental group of a CW-complex and give some applications of this theorem. Sometimes, it is more convenient to calculate fundamental groups by using exact sequences of bundles. Such is the case for, e.g., the fundamental group of SO(n). In many situations, the van Kampen theorem about the structure of the fundamental group of a union of two open sets is helpful. For example, it can be used to calculate the fundamental group of a knot complement. At the end of the chapter, we give another theorem of van Kampen, which gives a method for calculating the fundamental group of the complement of an algebraic curve in CP 2 . The corresponding calculations for particular curves are fairly complicated plenty of interesting results have been obtained, but many things are not yet fully understood. One of the main purposes of this book is to advance in the study of the properties of topological spaces (especially manifolds) as far as possible without employing complicated techniques. This distinguishes it from the majority of topology books.

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