A criterion for topological equivalence of two variable complex analytic function germs

We show that two analytic function germs (\C^2,0) \to (\C,0) are topologically right equivalent if and only if there is a one-to-one correspondence between the irreducible components of their zero sets that preserves the multiplicites of these components, their Puiseux pairs, and the intersection numbers of any pairs of distinct components.Comment: 6 page

Lifting differentiable curves from orbit spaces

Let $\rho : G \rightarrow \operatorname{O}(V)$ be a real finite dimensional orthogonal representation of a compact Lie group, let $\sigma = (\sigma_1,\ldots,\sigma_n) : V \to \mathbb R^n$, where $\sigma_1,\ldots,\sigma_n$ form a minimal system of homogeneous generators of the $G$-invariant polynomials on $V$, and set $d = \max_i \operatorname{deg} \sigma_i$. We prove that for each $C^{d-1,1}$-curve $c$ in $\sigma(V) \subseteq \mathbb R^n$ there exits a locally Lipschitz lift over $\sigma$, i.e., a locally Lipschitz curve $\overline c$ in $V$ so that $c = \sigma \circ \overline c$, and we obtain explicit bounds for the Lipschitz constant of $\overline c$ in terms of $c$. Moreover, we show that each $C^d$-curve in $\sigma(V)$ admits a $C^1$-lift. For finite groups $G$ we deduce a multivariable version and some further results.Comment: 25 pages; section on orbit spaces as differentiable spaces added, some typos corrected; accepted for publication in Transformation Group

Motivic-type Invariants of Blow-analytic Equivalence

To a given analytic function germ $f:(\mathbb{R}^d,0) \to (\mathbb{R},0)$, we associate zeta functions $Z_{f,+}$, $Z_{f,-} \in \mathbb{Z} [[T]]$, defined analogously to the motivic zeta functions of Denef and Loeser. We show that our zeta functions are rational and that they are invariants of the blow-analytic equivalence in the sense of Kuo. Then we use them together with the Fukui invariant to classify the blow-analytic equivalence classes of Brieskorn polynomials of two variables. Except special series of singularities our method classifies as well the blow-analytic equivalence classes of Brieskorn polynomials of three variables.Comment: 36 pages, 3 figure