This book is an introduction to manifolds at the beginning graduate
level. It contains the essential topological ideas that are needed for
the further study of manifolds, particularly in the context of
di?erential geometry, algebraic topology, and related ?elds. Its
guiding philosophy is to develop these ideas rigorously but
economically, with minimal prerequisites and plenty of geometric
intuition. Here at the University of Washington, for example, this
text is used for the ?rst third of a year-long course on the geometry
and topology of manifolds; the remaining two-thirds focuses on smooth
manifolds.
Therearemanysuperbtextsongeneralandalgebraictopologyavailable. Why add
another one to the catalog? The answer lies in my particular
visionofgraduateeducation—itismy(admittedlybiased)beliefthatevery
serious student of mathematics needs to know manifolds intimately, in
the same way that most students come to know the integers, the real
numbers, Euclidean spaces, groups, rings, and ?elds. Manifolds play a
role in nearly every major branch of mathematics (as I illustrate in
Chapter 1), and specialists in many ?elds ?nd themselves using
concepts and terminology fromtopologyandmanifoldtheoryonadailybasis.
Manifoldsarethuspart of the basic vocabulary of mathematics, and need
to be part of the basic graduate education. The ?rst steps must be
topological, and are embodied in this book; in most cases, they should
be complemented by material on smooth manifolds, vector ?elds,
di?erential forms, and the like. (After all, few of the really
interesting applications of manifold theory are possible without using
tools from calculus.
Les mer
Produktdetaljer
ISBN
9780387227276
Publisert
2020
Utgiver
Vendor
Springer
Språk
Product language
Engelsk
Format
Product format
Digital bok
Forfatter