The book "Foundations of Differentiable Manifolds and Lie Groups" by Frank W. Warner is a graduate-level textbook that provides a foundation for students interested in areas of mathematics requiring the concept of a differentiable manifold. It assumes a strong undergraduate background in algebra and analysis, along with some knowledge of point set topology, covering spaces, and the fundamental group. The text is also intended as a reference, including proofs of key theorems such as Hodge and de Rham. The core material is in Chapters 1, 2, and 4, covering differentiable manifolds, tangent vectors, submanifolds, implicit function theorems, vector fields, distributions, differential forms, integration, Stokes' theorem, and de Rham cohomology. Chapter 3 introduces Lie group theory, including the relationship between Lie groups and their Lie algebras, the exponential map, adjoint representation, and the closed subgroup theorem. It also discusses homogeneous manifolds. Chapter 5 develops axiomatic sheaf cohomology theory, proving the de Rham theorem, which shows that the de Rham cohomology ring is isomorphic to the differentiable singular cohomology ring. It also proves canonical isomorphisms of all classical cohomology theories on manifolds. Chapter 6 presents a complete treatment of the local theory of elliptic operators, using Fourier series as the basic tool, and includes the Hodge theorem. The book includes exercises at the end of each chapter, many of which are significant extensions of the text or contain major theorems. Some exercises provide hints for difficult problems. The text omits complex manifolds, infinite-dimensional manifolds, Sard's theorem, and imbedding theorems, referring readers to other texts for these topics. The book is suitable for a one-semester course covering the core material of Chapters 1, 2, and 4, with additional chapters depending on the class's interests. It can also be covered in a one-year course. The text is recommended for further study in differential geometry, with references to advanced texts provided. Warner thanks colleagues and students for their contributions and acknowledges Springer for republishing the text in the Graduate Texts in Mathematics series. The Springer edition corrects a few mathematical and typographical errors and adds additional titles to the bibliography.The book "Foundations of Differentiable Manifolds and Lie Groups" by Frank W. Warner is a graduate-level textbook that provides a foundation for students interested in areas of mathematics requiring the concept of a differentiable manifold. It assumes a strong undergraduate background in algebra and analysis, along with some knowledge of point set topology, covering spaces, and the fundamental group. The text is also intended as a reference, including proofs of key theorems such as Hodge and de Rham. The core material is in Chapters 1, 2, and 4, covering differentiable manifolds, tangent vectors, submanifolds, implicit function theorems, vector fields, distributions, differential forms, integration, Stokes' theorem, and de Rham cohomology. Chapter 3 introduces Lie group theory, including the relationship between Lie groups and their Lie algebras, the exponential map, adjoint representation, and the closed subgroup theorem. It also discusses homogeneous manifolds. Chapter 5 develops axiomatic sheaf cohomology theory, proving the de Rham theorem, which shows that the de Rham cohomology ring is isomorphic to the differentiable singular cohomology ring. It also proves canonical isomorphisms of all classical cohomology theories on manifolds. Chapter 6 presents a complete treatment of the local theory of elliptic operators, using Fourier series as the basic tool, and includes the Hodge theorem. The book includes exercises at the end of each chapter, many of which are significant extensions of the text or contain major theorems. Some exercises provide hints for difficult problems. The text omits complex manifolds, infinite-dimensional manifolds, Sard's theorem, and imbedding theorems, referring readers to other texts for these topics. The book is suitable for a one-semester course covering the core material of Chapters 1, 2, and 4, with additional chapters depending on the class's interests. It can also be covered in a one-year course. The text is recommended for further study in differential geometry, with references to advanced texts provided. Warner thanks colleagues and students for their contributions and acknowledges Springer for republishing the text in the Graduate Texts in Mathematics series. The Springer edition corrects a few mathematical and typographical errors and adds additional titles to the bibliography.