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.
