Topology of Infinite-Dimensional Manifolds

Abstract

This series publishes advanced monographs giving well-written presentations of the "state-of-the-art" in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

The strong universality of ANRs with a suitable algebraic structure

“…A topological space X is called an absolute (neighborhood) retract if X is metrizable and X is a (neighborhood) retract in each metrizable space containing X as a closed subspace. Absolute neighborhood retracts (briefly, ANRs) play an important role in Geometric and Infinite-Dimensional Topology [22], [23].…”

Characterizations of manifolds modeled on absorbing sets in non-separable Hilbert spaces and the discrete cells property

“…space is called σ-locally separable. Infinite-dimensional manifolds modeled on the product spaces 2 (κ) × f 2 , f 2 (κ) × 2 and f 2 (κ) × I ω are characterized as follows [9,13,6,12] ( 2 ):…”

“…Corollary 5.6(i) of [4] and Theorem 3.9(1) of [13] hold without some conditions on classes. The following proposition is proven in [12] ( 5 ). For the sake of completeness, we give its proof.…”

“…( 5 ) In [4,13], Proposition 5.2 is shown by applying the Triangulation Theorem for Hilbert manifolds under the assumption that C is I-stable. On the other hand, using the Open Embedding Theorem for Hilbert manifolds, K. Sakai proved it without I-stability in [12].…”

Branching geodesics of the Gromov--Hausdorff distance

“…The set K is a compact convex infinite-dimensional subset of the locally convex topological linear space R N . By the Dugundji extension theorem (see [14,Theorem 1.13.1]), K is an AR. Thus, by Theorem 5.8, the set K is homeomorphic to Q.…”