orse function is a Morse function such that the following hold ([@kitazawa]).

1.  At distinct singular points the values are distinct.

2.  Inverse images of regular values are disjoint unions of standard spheres.

3.  A vertex of the Reeb graph such that the inverse image includes a singular point not giving a local extremum is a vertex of degree $3$.

We consider pseudo quotient maps of the class of such functions. We regard these functions as $C^{\infty}$ functions whose measure zero sets are empty.

We present local forms of these pseudo quotient maps with several inverse images in FIGURE \[fig:5\].

![Local forms of the pseudo quotient maps with several inverse images.[]{data-label="fig:5"}](sphelocal.eps){width="30mm"}

We investigate the local form around a vertex of degree $3$ of a pseudo quotient map of the class of the functions with the inverse image. Consider a small regular neighborhood of the just one non-manifold point in the inverse image of the vertex. See FIGURE \[fig:6\].

![The local form around a vertex of degree $3$ of a pseudo quotient map of the class of the functions with the inverse image (a regular neighborhood of the just one non-manifold point in the inverse image of the vertex).[]{data-label="fig:6"}](localspheind.eps){width="15mm"}

If we remove the inverse image of the vertex, then the resulting space has $4$ connected components: two of them are in the upper part and the others are in the lower part. Moreover, the former two components are mapped onto an interval and the latter two ones are mapped onto the disjoint union of two intervals in the graph.

This leads us to the fact that a pseudo quotient map whose target is as FIGURE \[fig:7\] cannot be realized as a quotient map of the class. Arrows indicate natural local orientations induced from canonically obtained local functions. Note that around a vertex of degree $3$, the local map cannot be regared as a $D_3$-symmetric map.

![The graph represents the target of a pseudo quotient map of the class of standard-spherical functio