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Fold maps are smooth maps at each singular point of which it is representedas the product map of a Morse function and the identity map.  Round fold maps are, in short, such maps the sets of all singular points ofwhich are embedded concentrically. They are, as Morse functions, important inunderstanding the topologies and the differentiable structures in geometricways.  In the present paper, we study cohomology rings of $3$-dimensional manifoldsadmitting round fold maps into the plane and see that difference of thecoefficient rings and topological types of round fold maps are closely related.This is an explicit precise new study on our previous study, showing that a$3$-dimensional closed and orientable manifold is a so-called {\it graphmanifold}, or a manifold obtained by gluing so-called circle bundles oversurfaces along tori, if and only if it admits a round fold map into the plane.We also show another exposition on classifications of graph manifolds admittingsuch maps whose topological types are of a certain simplest class.

Round fold maps on $3$-dimensional manifolds and their integral and rational cohomology rings