The Complexity of Proper Homotopy Equivalence of Graphs

TL;DR

This paper proves that the classification of locally finite graphs up to proper homotopy equivalence is as complex as possible in the Borel hierarchy, and identifies a large class of graphs sharing a common equivalence class.

Contribution

It establishes Borel completeness for proper homotopy equivalence of graphs and finds a comeager class among infinite graphs, extending results to noncompact surfaces.

Findings

Proper homotopy equivalence is Borel complete for locally finite graphs.

There exists a comeager equivalence class among infinite graphs.

Results extend to homeomorphism of noncompact surfaces with pants decompositions.

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.