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

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.