An infinite class of bounded 4-manifolds having regular genus three

In the present paper the classification of PL 4-manifolds by means of the combinatorial invariant “regular genus” is proved to be not finite to one: indeed, the set of all $D^2$-bundles over $S^2$ (i.e. every bundle $\csi_c$ with Euler class $c$ and boundary L(c,1), $c \in Z-\{0,-1,-1}$, together with the trivial bundle $S^2 X D^2$) constitutes an infinite family of PL 4-manifolds with the same regular genus (equal to three). Further, general results are obtained, concerning PL 4-manifolds with “restricted gap” between their regular genus and the rank of their fundamental group, especially in case of free fundamental group.