Differential Topology

...there are reasons enough to warrant a coherent treatment of the main body of differential topology in the realm of Banach manifolds, which is at the same time correct and complete. This book fills the gap: whenever possible the manifolds treated are Banach manifolds with corners. Corners add to the complications and the authors have carefully fathomed the validity of all main results at corners. Even in finite dimensions some results at corners are more complete and better thought out here than elsewhere in the literature. The proofs are correct and with all details. I see this book as a reliable monograph of a well-defined subject; the possibility to fall back to it adds to the feeling of security when climbing in the more dangerous realms of infinite dimensional differential geometry.
Peter W. Michor

Inhalt

Real Differentiable Manifolds with Corners. The Differential of Maps over Open Sets of Quadrants of Banach Spaces. Differentiable Manifolds with Corners. Differentiable Maps. Topological Properties of the Differentiable Manifolds. Differentiable Partitions of Unity. Tangent Space of a Manifold at a Point. The Whitney Extension Theorem and the Inverse Mapping Theorem for Differentiable Manifolds with Corners. The Whitney Extension Theorem. The Inverse Mapping Theorem. Local Diffeomorphisms. Products of Differentiable Manifolds. Sum of Manifolds. Submanifolds and Immersions. Submanifolds. Immersions. Embeddings.Submersions and Quotient Manifolds. Submersions. Inverse Image of a Submanifold by Means of a Submersion. Regular Equivalence Relations. Quotient Manifolds. Subimmersions. Subimmersions. Manifolds of Germs of Submanifolds. Lie Groups. Lie Groups. Homogeneous Spaces and Orbits. Transversality. Transversal Map to a Submanifold. Transversal Family of Maps. Fibered Product of Manifolds. Transversal Submanifolds. Parametrized Theorems of the Density of the Transversality. Lebesgue Measure Zero Sets in Rm, m greater than or equal to 1. Subsets of Measure Zero in Finite Dimensional Manifolds. The Sard and Brown Theorems. Smale's and Quinn's density Theorems. Parametrized Theorem of the Density of the Transversality. Spaces of Differentiable Maps. Finite Order Jets between Differentiable Manifolds. Spaces of Continuous Maps. Topologies over the Spaces of Maps of Class p. Finite Order Whitney Topology. Infinite Order Jets. Whitney Topology of Infinite Order. Special Open Sets in the Spaces of Differentiable Maps. Continuity of the Composition of Differentiable Maps. Approximation of Differentiable Maps. Approximation of Differentiable Maps in the Case Finite Dimensional. Elevation of the Class of a Differentiable Manifold. Openness and Density of the Transversality. Density of the Transversality. R. Thom Theorem. Multijets. Density of the Transversality. Mather Theorems for Manifolds with Corners. Whitney Immersion Theorems. Morse Functions. Bibliography. Index.