Description: Manifolds are abound in mathematics and physics, and increasingly in cybernetics and visualization where they often reflect properties of complex systems and their configurations. Differential topology gives us the tools to study these spaces and extract information about the underlying systems. This book offers a concise and modern introduction to the core topics of differential topology for advanced undergraduates and beginning graduate students. It covers the basics on smooth manifolds and their tangent spaces before moving on to regular values and transversality, smooth flows and differential equations on manifolds, and the theory of vector bundles and locally trivial fibrations. The final chapter gives examples of local-to-global properties, a short introduction to Morse theory and a proof of Ehresmann's fibration theorem. The treatment is hands-on, including many concrete examples and exercises woven into the text, with hints provided to guide the student.
Brief description: Bjørn Ian Dundas is Professor in the Mathematics Department at the Universitetet i Bergen, Norway. Besides his research and teaching, he is the author of three books.
Review Quotes: 'For such studies, the present book is excellent. It scores on a number of counts: It does a solid job on the big topics that launch the subject i.e., manifolds, the tangent space, the cotangent space - the usual suspects, as Claude Rains would have it (also of differential geometry) ... The book starts with some marvelous and - at least to me - unexpected motivations, to wit, a discussion of how a robot's arm operating in 3-space sweeps out surprising manifolds (like the torus), a discussion of the configuration space of a pair of electrons, and a discussion of state spaces and fibre bundles ... You gotta love it. It looks like a very good book.' Michael Berg, MAA Reviews