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.

ALGEBRAIC TOPOLOGY: An Introduction starts with the combinatorial definition of simplicial (co) homology and its main properties (including duality for homology manifolds). Then the geometrical facet of (co) homology via bordism theory is sketched and it is shown that the corresponding theory for pseudomanifolds coincides with the homology obtained from the singular chain complex. The classical applications of (co) homology theory are included. Degree and fixed-point theory are presented with their extensions to infinite dimensional spaces. The book also contains a geometric approach to the Hurewicz theorem relating homology and homotopy. The last chapter exploits the algebraic invariants introduced in the book to give in detail the homotopical classification of the three-dimensional lens spaces. Each chapter concludes with a generous list of exercises and problems; many of them contain hints for their solution. Some groups of problems introduce a topic not included in the basic core of the book.

In essence the proceedings of the 1967 meeting in Baton Rouge, the volume offers significant papers in the topology of infinite dimensional linear spaces, fixed point theory in infinite dimensional spaces, infinite dimensional differential topology, and infinite dimensional pointset topology. Later results of the contributors underscore the basic soundness of this selection, which includes survey and expository papers, as well as reports of continuing research.

Geometric Topology contains the proceedings of the 1977 Georgia Topology Conference, held at the University of Georgia on August 1977. The book is comprised of contributions from leading experts in the field of geometric topology.These contributions are grouped into four sections: low dimensional manifolds, topology of manifolds, shape theory and infinite dimensional topology, and miscellaneous problems. Subjects discussed under these sections include local spanning missing loops, the structure of generalized manifolds having nonmanifold set of trivial dimension, universal open principal fibrations, and how to build a flexible polyhedral surface. Topologists, geometers, and mathematicians will find the book very interesting and insightful.