2020/03/24 by R. Giménez Conejero, Conejero, R. Giménez, J. J. Nuño‐Ballesteros +1 · 1 citation
Mathematics · #58K40 #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals #Primary 58K15 #Secondary 32S30
paper · pdf · doi:10.48550/arxiv.2003.10795
openalex publication_date 2020/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We show three basic properties on the image Milnor number μI(f) of a germ f\colon(ℂn,S)→(ℂn+1,0) with isolated instability. First, we show the conservation of the image Milnor number, from which one can deduce the upper semi-continuity and the topological invariance for families. Second, we prove the weak Mond's conjecture, which says that μI(f)=0 if and only if f is stable. Finally, we show a conjecture by Houston that any family ft\colon(ℂn,S)→(ℂn+1,0) with μI(ft) constant is excellent in Gaffney's sense. By technical reasons, in the two last properties we consider only the corank 1 case.