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arXiv · 0904.1197

On multiplicity of mappings between surfaces

Semeon Bogatyi·Jan Fricke·Elena Kudryavtseva
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Abstract

Let M and N be two closed (not necessarily orientable) surfaces, and f a continuous map from M to N. 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 in N under g is at most k. We calculate MMR[f] for any map $f$ of positive absolute degree A(f). The answer is formulated in terms of A(f), [pi_1(N):f_#(pi_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.

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Semeon Bogatyi, Jan Fricke, Elena Kudryavtseva. 2009-04-07. On multiplicity of mappings between surfaces. https://doi.org/10.2140/gtm.2008.14.49

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