Apparent Contours of Stable Maps into the Sphere

@article{Fukuda2011ApparentCO,
  title={Apparent Contours of Stable Maps into the Sphere},
  author={Taishi Fukuda and Takahiro Yamamoto},
  journal={Journal of Singing},
  year={2011},
  volume={3},
  pages={113-125}
}
For a stable map φ : M → S2 of a closed and connected surface into the sphere, let c(φ) and n(φ) denote the numbers of cusps and nodes respectively. In this paper, for each integer i ≥ 1, in the given homotopy class with i fold curve components, we will determine the minimal number c + n. 
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