Graphs and Surfaces: A Tale of Two Topologies

Sanghoon Kwak ()
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Sanghoon Kwak: Seoul National University, Department of Mathematics Education

A chapter in Handbook of Visual, Experimental and Computational Mathematics, 2026, pp 329-355 from Springer

Abstract: Abstract This chapter develops a “dictionary” between graphs and surfaces by comparing how each object is classified up to topological deformation and how its symmetry group is defined. For a graph, we consider the group of homotopy equivalences on it as its symmetry group where we identify two maps if they are homotopic to each other. This is reinterpreted as the group of outer automorphisms of a free group. For a surface, its symmetry group is defined to be the group of homeomorphisms where we identify two maps if they are homotopic. The same definition is used for infinite-type surfaces. For infinite graphs however, ordinary homotopy equivalence is shown to behave too flexibly, motivating the use of proper homotopy equivalences instead. In fact, a proper homotopy equivalence provides a nice classification of locally finite infinite graphs, so leads to a definition of the symmetry group of a locally finite infinite graph as the group of proper homotopy equivalences, identifying two maps if they are properly homotopic. Finally, we highlight a convergence (“Dehn–Nielsen–Baer”-type theorem) and a divergence between the two worlds: While the mapping class groups of surfaces and finite graphs exhibit rigidity, infinite graphs can fail rigidity.

Keywords: Graph; Surface; Topology; Symmetry; Homeomorphism; Homotopy equivalence; Mapping class groups (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16368-4_89

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