The Complexity of Proper Homotopy Equivalence of Graphs

Abstract

We demonstrate that the proper homotopy equivalence relation for locally finite graphs is Borel complete. Furthermore, among the infinite graphs, there is a comeager equivalence class. As corollaries, we obtain the analogous results for the homeomorphism relation of noncompact surfaces with pants decompositions.

Added an appendix proving that the association of the endspace pair to a graph is Borel. 17 pages, 10 figures