PROPER HOMOTOPY CLASSIFICATION OF GRAPHS R. AYALA, E. DOMINGUEZ, A. MARQUEZ AND A. QUINTERO ABSTRACT This work presents a classification of the proper homotopy types of locally finite 1-dimensional CW- complexes. 0. Introduction A proper map (p-map) is a continuous map/: X-> Y such that f~\K) is compact for any compact K. Proper homotopy (p-homotopy), proper homotopy equivalence, etc., are defined in the natural way. A graph is the underlying space of a connected locally finite 1-dimensional CW-complex. We consider on the half-line IR + the natural CW-structure whose 0-cells are the set of positive integers. Any cellular embedding f:U + -*G is called an infinite branch of the graph G. The vertex /[0) is called the root vertex of the branch. It is a well-known fact that any compact graph has the same ordinary homotopy type as a wedge of finitely many copies of the 1-sphere S 1 . We call that number the genus of the graph. So the genus classifies the ordinary homotopy type of compact graphs and actually gives us an equivalence with the category offinitelygenerated free groups. Nevertheless, there are easy examples of non-compact graphs with the same ordinary homotopy type but different p-homotopy types. The aim of this work is to state the corresponding proper homotopy classification of non-compact graphs. This problem was posed to us by Professor H. J. Baues as the starting point for the study of proper homotopy from a combinatorial point of view. As in the case of open surfaces (see [6]), the notion of Freudenthal end is the main tool used for obtaining such a classification. A Freudenthal end of a space X is an element of the inverse limit ^(X) = \imn Q (X— K), where Granges over the family of compact sets of X and n o (X— K) stands for the set of connected components. When A' is a 7^-locally compact cr-compact space, we can use a countable sequence K x c K 2 c ... of compact subsets to obtain !F{X). The topology of X can be enlarged to a topology on X = X U ^(X) in such a way that ^(X) turns out to be homeomorphic to a closed set of the Cantor set (see [3] and [1]), as follows. The topology on X is generated by the topology of X and the sets U = U\J U*, where Uen o (X—K n ) for some n and U* is the set of ends given by the sequences { U n } e lim n o (X— K n ) such that there is a positive integer « 0 with U n a U. Another useful proper invariant is the notion of proper end. A proper end is a p- homotopy class of p-maps/: U + -*• X. The set of proper ends of Xis denoted by F(X) and there is a map 0 from F(X) onto Received 11 September 1989; revised 8 December 1989. 1980 Mathematics Subject Classification 05CXX, 54C10, 55P15. Bull. London Math. Soc. 22 (1990) 417-421 15 BLM 22