On multiplicity of mappings between surfaces
Geometric Topology
2009-04-08 v1
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 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.
Cite
@article{arxiv.0904.1197,
title = {On multiplicity of mappings between surfaces},
author = {Semeon Bogatyi and Jan Fricke and Elena Kudryavtseva},
journal= {arXiv preprint arXiv:0904.1197},
year = {2009}
}
Comments
This is the version published by Geometry & Topology Monographs on 29 April 2008