Manifolds modelled on $R^{\infty }$ 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 | References | Similar Articles | Additional Information

Abstract: Let $R^\infty = \lim _ \to R^n$, 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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Keywords:
Bounded weak-<IMG WIDTH="16" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$\ast$"> topology,
manifold,
Banach space,
topological vector space,
direct limit,
Hilbert cube,
microbundle,
bundle,
stable classification

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© Copyright 1975
American Mathematical Society