On topologies of triangulated infinite-dimensional manifolds
Katsuro Sakai · Journal of the Mathematical Society of Japan · 1987
Consider $R^{n}$ as the subset $R^{n}\cross\{(0,0, \cdots)\}$ of the countable infinite product $R^{\omega}$ of the real line $R$ .The set $\bigcup_{n\in N}R^{n}$ admits two different natural topologies.One is the weak topology with respect to the tower $\{R^{n}\}_{n\in N}$ and the space with this topology is called the direct limit of lines and denoted by dir $limR^{n}$ or simply by $R^{\infty}$ .Another is the relative topology inherited from the product topology of $R^{\omega}$ and the space with this topology is denoted byis the linear span of the canonical orthonormal basis of Hilbert space $l_{2}.$ ) A separable topological manifold modeled on these spaces is called an $R^{\infty}$ -manifold or a $\sigma$ -manifold, respectively.These are considered as two different topologi- zations on the same underlying set.The former is the direct limit of a tower of finite-dimensional $(f.d.)$ compact metrizable spaces (compacta), that is, its topology is the weak topology with respect to the tower ([8, Prop.$m$ .$2]$ ).The latter is metrizable and coarser than the former.Both of these manifolds are triangulated, that is, each $R^{\infty}$ -manifold is homeomorphic to a simplicial complex with the weak (Whitehead) topology (cf.[18, Introduction]) and each $\sigma$ -manifold is homeomorphic to a simplicial complex with the metric topology ([11, Theorem 15]).Let $K$ be a simplicial complex and $|K|=\cup K$ the realization of $K$ .By $|K|_{w}$ and $|K|_{m}$ , we denote the spaces $|K|$ with the weak topology and the metric topology, respectively.We conjecture that $|K|_{w}$ is an $R^{\infty}$ -manifold if and only if $|K|_{m}$ is $a$ a-manifold.In this paper, we prove a half of this conjecture, that is,A map $f:Xarrow Y$ is a fine homotopy equivalence provided for each open cover $\mathcal{U}$ of $Y$ there exists a map $g:Yarrow X$ such that $fg$ is $\mathcal{U}$ -homotopic to $id_{Y}$ and