Hiroshima Math. J. 36 (2006), 175–202 Proper Lusternik-Schnirelmann p1-categories

Abstract. We define new proper homotopy invariants, the proper Lusternik-Schnirelmann p1-categories p~p1-cat and p~p y 1-cat. Then, we prove that, if p~p1-cat (resp. p~py1-cat) of a locally path-connected, Hausdor¤, locally compact, and paracompact space is equal to or less than n, then there is a proper map to a locally finite polyhedron of dimension nþ 1 that induces an isomorphism of fundamental pro-groups p~p1 (resp. p~py1)