Non-metrizable manifolds and contractibility

Abstract

We investigate whether non-metrizable manifolds in various classes can be homotopy equivalent to a CW-complex (in short: heCWc), and in particular contractible. We show that a non-metrizable manifold cannot be heCWc if it has one of the following properties: it contains a countably compact non-compact subspace; it contains a copy of an $\omega_1$-compact subset of an $\omega_1$-tree; it contains a non-Lindel\"of closed subspace functionally narrow in it. (These results hold for more general spaces than just manifolds.) We also show that the positive part of the tangent bundle of the long ray is not heCWc (for any smoothing). These theorems follow from stabilization properties of real valued maps. On a more geometric side, we also show that the Pr\"ufer surface, which has been shown to be contractible long ago, has an open submanifold which is not heCWc. On the other end of the spectrum, we show that there is a non-metrizable contractible Type I surface.