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Topology of Infinite-Dimensional Manifolds (Paperback, 1st ed. 2020)
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Topology of Infinite-Dimensional Manifolds (Paperback, 1st ed. 2020)
Series: Springer Monographs in Mathematics
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An infinite-dimensional manifold is a topological manifold modeled
on some infinite-dimensional homogeneous space called a model
space. In this book, the following spaces are considered model
spaces: Hilbert space (or non-separable Hilbert spaces), the
Hilbert cube, dense subspaces of Hilbert spaces being universal
spaces for absolute Borel spaces, the direct limit of Euclidean
spaces, and the direct limit of Hilbert cubes (which is
homeomorphic to the dual of a separable infinite-dimensional Banach
space with bounded weak-star topology). This book is designed for
graduate students to acquire knowledge of fundamental results on
infinite-dimensional manifolds and their characterizations. To read
and understand this book, some background is required even for
senior graduate students in topology, but that background knowledge
is minimized and is listed in the first chapter so that references
can easily be found. Almost all necessary background information is
found in Geometric Aspects of General Topology, the author's first
book. Many kinds of hyperspaces and function spaces are
investigated in various branches of mathematics, which are mostly
infinite-dimensional. Among them, many examples of
infinite-dimensional manifolds have been found. For researchers
studying such objects, this book will be very helpful. As
outstanding applications of Hilbert cube manifolds, the book
contains proofs of the topological invariance of Whitehead torsion
and Borsuk's conjecture on the homotopy type of compact ANRs. This
is also the first book that presents combinatorial -manifolds, the
infinite-dimensional version of combinatorial n-manifolds, and
proofs of two remarkable results, that is, any triangulation of
each manifold modeled on the direct limit of Euclidean spaces is a
combinatorial -manifold and the Hauptvermutung for them is true.
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