Infinite products which are Hilbert cubes

Abstract:

Let $Q$ denote the Hilbert cube. It is shown that if $P$ and $P’$ are compact polyhedra of the same simple homotopy type then $P \times Q$ and $P’ \times Q$ are homeomorphic. A consequence of this result is that the Cartesian product of a countable, locally finite simplicial complex with a separable, infinite-dimensional Fréchet space is a manifold modelled on the Fréchet space. It is also proved that a countably infinite product of nondegenerate spaces is a Hilbert cube provided that the product of each of the spaces with the Hilbert cube is a Hilbert cube. Together with the first result, this establishes that a countably infinite product of nondegenerate, compact, contractible polyhedra is a Hilbert cube. In addition, a proof is given of the (previously unpublished) theorem of R. D. Anderson that a countably infinite product of nondegenerate dendra is a Hilbert cube.