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

Abstract

Let R = lim Rn, and let B*(b*) denote the conjugate, B*, of a separable, infinite-dimensional Banach space with its bounded weak-* topology. We investigate properties of paracompact, topological manifolds M, N modelled on F, where F is either R. or B*(b*). Included among our results are that locally trivial bundles and microbundles over M with fiber F are trivial; there is an open embedding M M X F; and if M and N have the same homotopy type, then M X F and N X F are homeomorphic. Also, if U is an open subset of B*(b*), then U X B*(b*) is homeomorphic to U. Thus, two open subsets of B*(b*) are homeomorphic if and only if they have the same homotopy type. Our theorems about B*(b*)-manifolds, B*(b*) 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. 0. Introduction. Let R' = lim Rn, and let B*(b*) denote the conjugate, B*, of a separable, infinite-dimensional Banach space with its bounded weak-* topology. The bounded weak-* topology is the finest topology agreeing with the weak-* topology on bounded sets. (The weak-* topology on B* is the smallest topology on which all the linear functionals {xlIx E B} are continuous where A(a) = a(x).) We investigate properties of paracompact, topological manifolds, M, N, modelled on F, where F is either R' or B*(b*). Included among our results are that locally trivial bundles and microbundles over M with fiber F are trivial; there is an open embedding M M x F; and if M and N have the same homotopy type, then M x F is homeomorphic to N x F. We also show that if U is open in B*(b*), then U x B*(b*) is homeomorphic to U. Thus, two open subsets of B*(b*) are homeomorphic if and only if they have the same homotopy type. Any theorem about B*(b*)-manifolds, B*(b*) as above, yields an analogous Received by the editors May 10, 1973 and, in revised form, April 17, 1974. AMS (MOS) subject classifications (1970). Primary 58B05; Secondary 46A99.