# zbMATH — the first resource for mathematics

The invariance of Milnor’s number implies the invariance of the topological type. (English) Zbl 0351.32009
Let $$F(t,z)$$ be a polynomial in $$z=(z_0,z_1,\dots,z_n)$$ with coefficients which are smooth complex valued functions of $$t\in I= [0,1]$$ such that $$F(t,0)=0$$ and that for each $$t\in I$$, the polynomials $$(\partial F/\partial z_i)(t,z)$$ in $$z$$ have an isolated zero at $$0$$. Assume more over that the integer $$\mu_t=\dim_{\mathbb{C}}\mathbb{C}/(\partial f/\partial z_0)(t,z),\dots,(\partial f/\partial z_n)(t,z))$$ is independent of $$t$$. The authors prove that the monodromy fibrations of the singularities of $$F(0,z) =0$$ and $$F(1, z) =0$$ at $$0$$ are of the same fiber homotopy and, if further $$n\neq 2$$, these fibrations are even differentially isomorphic and the topological types of the singularities are the same. The hypothesis $$n\neq 2$$ comes from using h-cobordism theorem. This gives a proof of the Hironaka’s conjecture for $$n=1$$ in the more general case of a $$C^{\infty}$$ family of $$n$$-dimensional hypersurfaces of dimension $$n\neq 2$$. Making use of the results of K. Brauner [Abh. math. Semin. Hamburg Univ. 6, 1–55 (1928; JFM 54.0373.01)], W. Burau [Abh. Math. Semin. Hamb. Univ. 9, 125–133 (1932; Zbl 0006.03402; JFM 58.0615.01)] and O. Zariski [Am. J. Math. 54, 453–465 (1932; Zbl 0004.36902; JFM 58.0614.02)], the authors prove that Puiseux pairs of an analytically irreducible plane curve singularity depends only on the topology of the singularity.

##### MSC:
 32Sxx Complex singularities 14B05 Singularities in algebraic geometry
Full Text: