  This note is the observation that a simple combination of known results shows
that the usual analytic invariants of a finitely determined multi-germ $f\colon
(\mathbb{C}^2,S)\to(\mathbb{C}^3,0)$ ---namely the image Milnor number $\mu_I$,
the number of crosscaps and triple points, $C$ and $T$, and the Milnor number
$\mu(\Sigma)$ of the curve of double points in the target--- depend only on the
embedded topological type of the image of $f$. As a consequence one obtains the
topological invariance of the sign-refined Smale invariant for immersions
$j\colon S^3\looparrowright S^5$ associated to finitely determined map germs
$(\mathbb{C}^2,0)\to(\mathbb{C}^3,0)$.
  This note is a corrected version of a previous homonymous work containing an
error. A previous wrong computation of $b_1(\mathbb{F})$, spotted by Siersma,
has been replaced by the correct statement, due to Van Straten. This has forced
the proofs for the mentioned topological invariances to differ significantly
from the previous version.
