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This book provides an introduction to prerequisite topological and algebraic knowledge. The first seven chapters, focusing on local analysis, can be used as a fundamental textbook for graduate students of theoretical physics. The later chapters discuss geometry in fibre bundles, and deal with the Atiyah-Singer index theorem, its generalization and its application, quantum anomaly, cohomology field theory and noncommutative geometry. The text aims to provide coverage of current research in theoretical physics.