Apparent contours of stable maps of surfaces with boundary into the plane

Takahiro Yamamoto

Journal of Singularities
volume 22 (2020), 114-133

Received: 25 February 2019. Received in revised form: 25 May 2020.

DOI: 10.5427/jsing.2020.22h

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Abstract:

Let M be a connected compact surface with boundary. A C^∞ map M -> R^2 is admissible if it is non-singular on a neighborhood of the boundary. For a C^∞ stable map f: M -> R^2, denote by c(f) and n(f), i(f) the number of cusps and nodes, connected components of the set of singular points respectively. In this paper, we introduce the notion of admissibly homotopic among C^∞ maps M -> R^2, and we will determine the minimal number c +n for each admissibly homotopy class.

2000 Mathematical Subject Classification:

Primary 57R45; Secondary 57R35, 57R90, 58K65

Key words and phrases:

stable map, cusps, nodes

Author(s) information:

Takahiro Yamamoto
Department of Mathematics
Tokyo Gakugei University
Koganei, Tokyo, 184-8501, Japan
email: yamush@u-gakugei.ac.jp