A 𝑄-manifold local-compactification of a metric combinatorial ∞-manifold

Abstract

Let K K be a combinatorial ∞ \infty -manifold, that is, a countable simplicial complex such that the star of each vertex is combinatorially equivalent to the countable-infinite full simplicial complex. Then the space | K | m {\left | K \right |_m} with the metric topology is a manifold modeled on the space σ \sigma , where σ \sigma is the subspace of the Hilbert cube Q = I ω Q = {I^\omega } which consists of all points having at most finitely many nonzero coordinates. In this paper, we give a local-compactification of | K | m {\left | K \right |_m} which is a [ 0 , 1 ) [0,1) -stable Q Q -manifold containing | K | m {\left | K \right |_m} as an f.d. cap set.