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On multiplicity of mappings between surfaces

Semeon Bogatyi, Jan Fricke and Elena Kudryavtseva

Geometry & Topology Monographs 14 (2008) 49–62

DOI: 10.2140/gtm.2008.14.49

arXiv: 0904.1197


Let M and N be two closed (not necessarily orientable) surfaces, and f:M → N a continuous map. By definition, the minimal multiplicity MMR[f] of the map f denotes the minimal integer k having the following property: f can be deformed into a map g such that the number ∣g-1(c)∣ of preimages of any point c∈ N under g is ≤ k. We calculate MMR[f] for any map f of positive absolute degree A(f). The answer is formulated in terms of A(f), [π1(N):f#1(M))], and the Euler characteristics of M and N. For a map f with A(f)=0, we prove the inequalities 2≤MMR[f]≤4.

In grateful memory of Heiner, his wonderful collaboration and friendship


multiplicity of a map, absolute degree, surface, homotopy, branched covering

Mathematical Subject Classification

Primary: 54H25

Secondary: 55M20, 57M12


Received: 29 April 2006
Accepted: 9 November 2006
Published: 29 April 2008

Semeon Bogatyi
Department of Mathematics and Mechanics
Moscow State University
Moscow 119992
Jan Fricke
Fachbereich 6 – Mathematik
Universität Siegen
57068 Siegen
Elena Kudryavtseva
Department of Mathematics and Mechanics
Moscow State University
Moscow 119992