  We show that given any germ of complex analytic function
$f\colon(X,x)\to(\mathbb{C},0)$ on a complex analytic space $X$, there exists a
geometric local monodromy without fixed points, provided that $f\in\mathfrak
m_{X,x}^2$, where $\mathfrak m_{X,x}$ is the maximal ideal of $\mathcal
O_{X,x}$. This result generalizes a well-known theorem of the second named
author when $X$ is smooth at $x$ and it also implies the A'Campo theorem that
the Lefschetz number of the monodromy is equal to zero. Moreover, we give an
application to the case that $X$ has maximal rectified homotopical depth at $x$
and show that a family of such functions with isolated critical points and
constant total Milnor number has no coalescing of singularities.
