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.
Author(s): Katsuro Sakai
Series: Springer Monographs in Mathematics
Edition: 1
Publisher: Springer
Year: 2020
Language: English
Pages: 619
Tags: Topology, Manifold, Banach Space, Hilbert Space, Cell Complex, Hilbert Cube, Triangulation,
Preface
Contents
1 Preliminaries and Background Results
1.1 Terminology and Notation
1.2 Banach Spaces in the Product of Real Lines
1.3 Topological Spaces
1.4 Linear Spaces and Convex Sets
1.5 Cell Complexes and Simplicial Complexes
1.6 Simplicial Subdivisions
1.7 The Metric Topology of Polyhedra
1.8 PL Maps and Simplicial Maps
1.9 Derived and Regular Neighborhoods
1.10 The Homotopy Type of Simplicial Complexes
1.11 The Nerves of Open Covers
1.12 Dimensions
1.13 Absolute Neighborhood Retracts
1.14 Locally Equi-Connected Spaces
1.15 Cell-Like Maps and Fine Homotopy Equivalences
Notes for Chapter 1
2 Fundamental Results on Infinite-Dimensional Manifolds
2.1 Remarks on the Model Spaces and Isotopies
2.2 The Toruńczyk Factor Theorem
2.3 Stability and Deficiency
2.4 Negligibility and Deficiency
2.5 The Collaring and Unknotting Theorems
2.6 Classification of E-Manifolds
2.7 The Bing Shrinking Criterion
2.8 Z-Sets and Strong Z-Sets in ANRs
2.9 Z-Sets and Strong Z-Sets in E-Manifolds
2.10 Z-Sets in the Hilbert Cube and Q-Manifolds
2.11 Complementary Basic Results on Q-Manifolds
Notes for Chapter 2
3 Characterizations of Hilbert Manifolds andHilbert Cube Manifolds
3.1 (Strong) Universality and U-Maps
3.2 The Discrete (or Disjoint) Cells Property
3.3 The Discrete F.D. Polyhedra Property
3.4 Characterization of 2(Γ)-Manifolds
3.5 Fréchet Spaces and the Countable Product of ARs
3.6 The Function Space C(X,Y)
3.7 Cell-Like Images of Q-Manifolds
3.8 Characterization and Classification of Q-Manifolds
3.9 Keller's Theorem and the Countable Product of ARs
Notes for Chapter 3
4 Triangulation of Hilbert Cube Manifolds and Related Topics
4.1 Simple Homotopy Equivalences
4.2 Covering Spaces and Algebraic Preliminaries
4.3 The Procedure for Killing Homotopy Groups
4.4 The Splitting Theorem for Q-Manifolds
4.5 An Immersion of a Punctured n-Torus into Rn
4.6 The Handle Straightening Theorem
4.7 The Triangulation Theorem for Q-Manifolds
4.8 Further Results on Q-Manifolds
Notes for Chapter 4
5 Manifolds Modeled on Homotopy Dense Subspacesof Hilbert Spaces
5.1 Cap Sets and F.D.Cap Sets in 2 and Q
5.2 Manifold Pairs Modeled on the Pair of 2 (or Q) andits (F.D.)Cap Set
5.3 Absorption Property and Absorption Bases
5.4 Absorption Bases and Strong Universality
5.5 Four Types of Absorption Bases for 2(Γ)
5.6 C-Universality and Cσ-Universality
5.7 Manifolds Modeled on Absorption Bases for 2(Γ)
5.8 Homotopy Dense Embedding into an 2(Γ)-Manifold
5.9 Four Types of Infinite-Dimensional Manifolds
5.10 Absorbing Sets
5.11 Absolute Borel Classes
5.12 Universal Spaces for the Absolute Borel Classes
Notes for Chapter 5
6 Manifolds Modeled on Direct Limits andCombinatorial ∞-Manifolds
6.1 Preliminaries for Direct Limits
6.2 The Bounded Weak-Star Topology
6.3 Embedding Neighborhood Extension Properties
6.4 Characterization of R∞- and Q∞-Manifolds
6.5 Classification of R∞- and Q∞-Manifolds
6.6 Simplicial Approximations of PL Embeddings
6.7 Retriangulating the Simplicial Neighborhood
6.8 Flattening Simplicial Subdivisions
6.9 Combinatorial ∞-Manifolds and PL ∞-Manifolds
6.10 Characterization of PL ∞-Manifolds
6.11 Characterization of Combinatorial ∞-Manifolds
6.12 Simplicial Complexes with Contractible Links
6.13 Metric Combinatorial ∞-Manifolds
6.14 Bi-topological Infinite-Dimensional Manifolds
Notes for Chapter 6
LF-Spaces
Piecewise Linear R∞-Manifolds
Products of Absorbing Sets and R∞
Appendix: PL n-Manifolds and Combinatorial n-Manifolds
A.1 Characterizations of Combinatorial n-Manifolds
A.2 The Boundary of a Combinatorial n-Manifold
A.3 Regular Neighborhoods in PL n-Manifolds
A.4 PL Embedding Approximation Theorem
Epilogue
Bibliography
Books and Texts
References
Index
