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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Manifolds modelled on $ R\sp{\infty }$ or bounded weak-* topologies


Author: Richard E. Heisey
Journal: Trans. Amer. Math. Soc. 206 (1975), 295-312
MSC: Primary 58B05; Secondary 58C20
DOI: https://doi.org/10.1090/S0002-9947-1975-0397768-X
MathSciNet review: 0397768
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Abstract: Let $ {R^\infty } = \mathop {\lim {R^n}}\limits_ \to $, and let $ {B^ \ast }({b^ \ast })$ denote the conjugate, $ {B^ \ast }$, of a separable, infinite-dimensional Banach space with its bounded weak-$ \ast $ topology. We investigate properties of paracompact, topological manifolds $ M,N$ modelled on $ F$, where $ F$ is either $ {R^\infty }$ or $ {B^ \ast }({b^ \ast })$. Included among our results are that locally trivial bundles and microbundles over $ M$ with fiber $ F$ are trivial; there is an open embedding $ M \to M \times F$; and if $ M$ and $ N$ have the same homotopy type, then $ M \times F$ and $ N \times F$ are homeomorphic. Also, if $ U$ is an open subset of $ {B^ \ast }({b^ \ast })$, then $ U \times {B^ \ast }({b^ \ast })$ is homeomorphic to $ U$. Thus, two open subsets of $ {B^ \ast }({b^ \ast })$ are homeomorphic if and only if they have the same homotopy type. Our theorems about $ {B^ \ast }({b^ \ast })$-manifolds, $ {B^ \ast }({b^ \ast })$ as above, immediately yield analogous theorems about $ B(b)$-manifolds, where $ B(b)$ is a separable, reflexive, infinite-dimensional Banach space with its bounded weak topology.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0397768-X
Keywords: Bounded weak-$ \ast $ topology, manifold, Banach space, topological vector space, direct limit, Hilbert cube, microbundle, bundle, stable classification
Article copyright: © Copyright 1975 American Mathematical Society