Simplicial Maps of the Complexes of Curves on Nonorientable Surfaces

2011/12/07 by Irmak, Elmas
#20F38 #57Mxx #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1112.1617

Abstract

Let N be a compact, connected, nonorientable surface of genus g with n boundary components. Let λ be a simplicial map of the complex of curves, C(N), on N which satisfies the following: [a] and [b] are connected by an edge in C(N) if and only if λ([a]) and λ([b]) are connected by an edge in C(N) for every pair of vertices [a], [b] in C(N). We prove that λ is induced by a homeomorphism of N if (g, n) ∈ \(1, 0), (1, 1), (2, 0), (2, 1), (3, 0)\ or g + n ≥ 5. Our result implies that superinjective simplicial maps and automorphisms of C(N) are induced by homeomorphisms of N.

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