Topology of Infinite-Dimensional Manifolds

verfasst von: Dr. Katsuro Sakai

Verlag: Springer Singapore

Buchreihe : Springer Monographs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries and Background Results

In this chapter, we first introduce terminology and notation, then list background results. The reader may skip this chapter and read the necessary parts later when needed. Several results might be learned in graduate courses, but others are advanced and special.

Katsuro Sakai

Chapter 2. Fundamental Results on Infinite-Dimensional Manifolds

In this chapter, we prove fundamental results on manifolds modeled on Hilbert space (more generally an infinite-dimensional normed linear space E such that E ≈ E ^N or $E\approx E_f^N$) or the Hilbert cube. We also prove the Toru’nczyk Factor Theorem, that is, for each complete metrizable ANR X with weight $\leqslant \tau $ ($\leqslant \aleph _0$), the product of X with the Hilbert space of weight τ is a Hilbert manifold.

Katsuro Sakai

Chapter 3. Characterizations of Hilbert Manifolds and Hilbert Cube Manifolds

Abstract

In this chapter, we prove the Toruńczyk characterizations of Hilbert manifolds (non-separable Hilbert spaces are also considered as model spaces) and Hilbert cube manifolds. By using the characterization of Hilbert space, we show that every Fréchet space is homeomorphic to Hilbert space with the same weight. Additionally, it is shown that if X is a non-discrete compactum and Y is a completely metrizable separable ANR Y without isolated points, then the space $\operatorname {C}(X,Y)$ of maps from X to Y is an ℓ ²-manifold. By using the characterization of Q, we prove Keller’s Theorem, that is, every infinite-dimensional compact metrizable convex set in a locally convex topological linear space is homeomorphic to Q.

Katsuro Sakai

Chapter 4. Triangulation of Hilbert Cube Manifolds and Related Topics

Abstract

In this chapter, we focus on proving the Triangulation Theorem for Hilbert cube manifolds. The compact case is bound up with the Borsuk conjecture asserting that every compact ANR has the homotopy type of a finite simplicial complex. The proof is based on Wall’s work on the homotopy type of a finite simplicial complex (Theorem 4.1.6) and a result on simple homotopy equivalences (Theorem 4.1.5). We follow the course of Chapman’s Lecture Notes (Lectures on Hilbert Cube Manifolds, CBMS Regional Conf. Ser. in Math. vol. 28, Amer. Math. Soc., Providence, 1975) and provide details for proofs. Combining the Triangulation Theorem with the Edwards Factor Theorem 3.8.1, we know that the Borsuk conjecture is true. We also prove the topological invariance of Whitehead torsion, which had been a longstanding problem.

Katsuro Sakai

Chapter 5. Manifolds Modeled on Homotopy Dense Subspaces of Hilbert Spaces

In this chapter, we introduce (f.d.)cap sets in the separable Hilbert space (or the Hilbert cube) and characterize manifolds modeled on them. Then, we discuss their non-separable version called absorption bases, which are absolutely F _σ and homeomorphic one of the following spaces:

$$\displaystyle \begin{aligned}\ell ^2_f(\varGamma ),\ \ell ^2_f(\varGamma )\times \ell ^2_Q,\ \ell ^2_f(\varGamma )\times \ell ^2, \text{ and }\ ; \ell ^2(\varGamma )\times \ell ^2_f, \end{aligned} $$

where Γ is an infinite set (if Γ is countable, then the last two spaces are the same). We characterize manifolds modeled on these spaces. Moreover, introducing absorbing sets, which are non-F _σ versions of cap sets or absorption bases, we characterize manifolds modeled on the universal spaces for absolute Borel classes, where non-separable cases are also discussed.

Katsuro Sakai

Chapter 6. Manifolds Modeled on Direct Limits and Combinatorial ∞-Manifolds

Abstract

Let

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and

https://static-content.springer.com/image/chp%3A10.1007%2F978-981-15-7575-4_6/369475_1_En_6_IEq2_HTML.gif

denote the direct limits of the following towers:

$$\displaystyle \mathbb {R} \subset \mathbb {R}^2 \subset \mathbb {R}^3 \subset \cdots \ ; \; \boldsymbol {Q} \subset \boldsymbol {Q}^2 \subset \boldsymbol {Q}^3 \subset \cdots . $$

As naturally expected, $\mathbb {R}^\infty $ is a locally convex topological linear space (Theorem 6.1.4). The space Q ^∞ is also homeomorphic to a locally convex topological linear space. In fact, it is known as the Heisey Theorem (Theorem 6.2.3) that Q ^∞ is homeomorphic to the dual space of any separable infinite-dimensional Banach space with the bounded weak-star topology. In this chapter, we give characterizations of $\mathbb {R}^\infty $-manifolds and Q ^∞-manifolds. Fundamental results on these manifolds are also demonstrated. An $\mathbb {R}^\infty $-manifold is triangulable. Namely, it is homeomorphic to the polyhedron of some simplicial complex. In the second half of this chapter, we will study triangulations of $\mathbb {R}^\infty $-manifolds. Generalizing combinatorial manifolds to the infinite-dimensional case, we introduce combinatorial ∞-manifolds. For these manifolds, the Hauptvermutung can be proved, though that does not hold for combinatorial n-manifolds if 3 < n < ∞. It is also proved that every triangulation of an $\mathbb {R}^\infty $-manifold is a combinatorial ∞-manifold.

Katsuro Sakai

Backmatter

Titel: Topology of Infinite-Dimensional Manifolds
verfasst von: Dr. Katsuro Sakai
Verlag: Springer Singapore
Electronic ISBN: 978-981-15-7575-4
Print ISBN: 978-981-15-7574-7
DOI: https://doi.org/10.1007/978-981-15-7575-4