Algebraic Topology is a comprehensive book that explores the fundamental concepts and principles of algebraic topology. Written by the renowned mathematician, Solomon Lefschetz, this book is part of the American Mathematical Society's Colloquium Publications series and is the 27th volume in the series. The book begins with an introduction to the basic concepts of topology, including the definition of a topological space, continuous maps, and homotopy. It then delves into the algebraic aspects of topology, exploring the fundamental group, homology, and cohomology. The book also covers a range of advanced topics, including fiber bundles, spectral sequences, and characteristic classes. Throughout the book, Lefschetz provides clear and concise explanations of complex mathematical concepts, making it accessible to both students and researchers. The book is filled with numerous examples and exercises to help readers deepen their understanding of the material. Overall, Algebraic Topology is an essential reference for anyone interested in the field of algebraic topology, including graduate students, researchers, and mathematicians.This scarce antiquarian book is a facsimile reprint of the old original and may contain some imperfections such as library marks and notations. Because we believe this work is culturally important, we have made it available as part of our commitment for protecting, preserving, and promoting the world's literature in affordable, high quality, modern editions, that are true to their original work.