  We give the definition of the Thom condition and we show that given any germ
of complex analytic function $f:(X,x)\to(\mathbb{C},0)$ 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 and proves a statement by
Tibar in his PhD thesis. 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.
