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Lectures on Differential Topology (Paperback)
Loot Price: R2,346
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Lectures on Differential Topology (Paperback)
Series: Graduate Studies in Mathematics
Expected to ship within 12 - 19 working days
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This book gives a comprehensive introduction to the theory of
smooth manifolds, maps, and fundamental associated structures with
an emphasis on ``bare hands'' approaches, combining
differential-topological cut-and-paste procedures and applications
of transversality. In particular, the smooth cobordism cup-product
is defined from scratch and used as the main tool in a variety of
settings. After establishing the fundamentals, the book proceeds to
a broad range of more advanced topics in differential topology,
including degree theory, the Poincare-Hopf index theorem,
bordism-characteristic numbers, and the Pontryagin-Thom
construction. Cobordism intersection forms are used to classify
compact surfaces; their quadratic enhancements are developed and
applied to studying the homotopy groups of spheres, the bordism
group of immersed surfaces in a 3-manifold, and congruences mod 16
for the signature of intersection forms of 4-manifolds. Other
topics include the high-dimensional $h$-cobordism theorem stressing
the role of the ``Whitney trick'', a determination of the singleton
bordism modules in low dimensions, and proofs of parallelizability
of orientable 3-manifolds and the Lickorish-Wallace theorem. Nash
manifolds and Nash's questions on the existence of real algebraic
models are also discussed. This book will be useful as a textbook
for beginning masters and doctoral students interested in
differential topology, who have finished a standard undergraduate
mathematics curriculum. It emphasizes an active learning approach,
and exercises are included within the text as part of the flow of
ideas. Experienced readers may use this book as a source of
alternative, constructive approaches to results commonly presented
in more advanced contexts with specialized techniques.
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