Proper L–S category, fundamental pro-groups and 2-dimensional proper co-H-spaces

This paper presents a study of one-ended locally finite CW-complexes with proper L–S category ⩽2. We detect the class of towers of groups which can be the fundamental pro-group of a space of proper L–S category 2. A second part of the paper is concerned with two-dimensional CW-complexes. For these, we give different characterizations of spaces with proper L–S category ⩽2.