Elements of Combinatorial and Differential Topology
The main topic of this book is combinatorial and differential topology. The author discusses a lot of interesting and basic facts avoiding sophisticated techniques, hence the reading of the book requires only a modest background for these topics (e.g. basic topological properties of sets in Euclidean space). After an introductory discussion of graphs, the topology of subsets in Euclidean space is considered (including the Jordan theorem for curves, the Brouwer fixed point theorem and the Sperner lemma). Simplicial complexes and CW-complexes are discussed in the next chapter, followed by a treatment of surfaces, coverings, fibrations and homotopy groups. The fifth chapter turns to differential topology (smooth manifolds, embeddings and immersions, the degree of a map, the Hopf theorem on the homotopy classification of maps to the sphere and Morse theory). The last chapter treats the fundamental groups (with many explicit examples). The book contains a lot of problems and their solutions can be found at the end of the book.