On the regular genus of 5-manifolds with free fundamental group

In the present paper, we obtain the following classification of closed orientable PL 5-manifolds M^5 with free fundamental group of rank m, so that the difference between the regular genus G(M^5) and m is less or equal to eight: (a) G(M^5) = m iff $M^5= \#_m (S^1 X S^4)$; (b) it is impossible m+1 less than or equal to G(M^5) less than or equal to m+7; (c) if G(M^5) = m+8, then either $M^5= \#_m (S^1 X S^4) # (S^2 X S^3)$ or $M^5= \#_m (S^1 X S^4) # (S^2 X_\sim S^3)$. As a consequence, we complete the classification of PL 5-manifolds up to regular genus eight, and compute the regular genus of the 5-dimensional real projective space RP5: if G(M^5) = 8, then either $M^5= S^2 X S^3$ or $M^5= S^2 X_\sim S^3$ or $M^5= \#_8 (S^1 X S^4)$; $G(S^2 X S^3) = 8$; $G(S^2 X_\sim S^3)$ greater than or equal to 8; $G(RP^5) = 9$.