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This book introduces the reader to the two most fundamental concepts of algebraic topology: the fundamental group and homology. We shall take a modern viewpoint so that we begin the course by studying basic notions from category theory. The fundamental group is afterwards treated as a special case of the fundamental groupoid. Accordingly, we first prove van Kampen’s theorem in a categorical version due to R. Brown and then explain how to actually compute the fundamental group of an attaching space. We move on to present homology. To convey the idea, we construct simplicial homology and motivate the Eilenberg–Steenrod axioms of a homology theory. Next, we construct singular homology and show that it satisfies the axioms. Afterwards, we develop machinery for computing homology theories, give example calculations, and see some applications such as the Brouwer fixed point theorem and the Borsuk–Ulam theorem. Finally, we introduce cellular homology for CW complexes. As the concluding result, we show that the Eilenberg Steenrod axioms determine ordinary homology on CW complexes. We assume the reader has taken an introductory course on topology and is familiar with point set topological concepts, the definition of the fundamental group, and covering theory. Such a course will certainly have covered the quotient topology, but since this concept is of fundamental importance for our purposes, we have decided to include a recap in Appendix A. Elementary algebra will likewise be applied throughout the text. In particular, we will work with modules over commutative rings and on rare occasions also with their tensor products. Each chapter ends with a couple of exercises. Some of them are not only meant to provide a test ground for working with the new concepts but they also establish additional facts and terminology which are helpful to know in the given context. A myriad of excellent books on algebraic topology are available in the market. Some texts, for example, [29], choose a formal presentation and are well suited to continue one’s curriculum with a course on homotopy theory. Others, like [8] are more geometrically minded and might be a better choice for subsequent specialization in low-dimensional topology. What this text tries to accomplish is to neither shy away from abstract concepts nor from providing geometric intuition or doing easy calculations, and at the same time do justice to the series and be a compact textbook: We present only as much material as we found reasonable to cover in a first semester graduate course on algebraic topology. The six chapters are divided into five sections each, so that in a typical 15-week semester with two meetings a week, one should cover one section per lecture on average. This is however a rough estimate as some sections are more substantial than others so that the blackboard presentation will need some shortcuts. However, we urge the lecturer not to try and skip the technical appearing Sect. 2.3 on cofibrations and homotopy pushouts. It secretly provides as much homotopy theory as we deem necessary for the correct presentation of the results in the ensuing chapters.