Classifying PL 5-manifolds by regular genus: the boundary case

In the present paper we face the problem of classifying classes of orientable PL 5-manifolds $M^5$ with $h\ge1$ boundary components, by making use of a combinatorial invariant called “regular genus” $G(M^5)$. In particular, a complete classification up to regular genus five is obtained: if $G(M^5)=\rho \le 5,$ then $M^5= \#_{\rho -\rho’} (S^1 x S^4) \# H_{\rho’}^h$where $\rho’=G(\partial M^5}$ denotes the regular genus of the boundary $\partial M^5$ and $H_{\rho’}^h$ denotes the connected sum of h orientable 5-dimensional handlebodies $Y_{\alpha_i}$ of genus $\alpha_i \ge 0$ so that $\sum_{i=1,…,h} \alpha_i = \rho’$. Moreover, we give a characterization of orientable PL 5-manifolds $M^5$ with boundary satisfying particular conditions related to the “gap” between $G(M^5)$ and either $ G(\partial M^5}$ or the rank of their fundamental group. Further, the paper explains how the above results (together with other known properties of regular genus of PL-manifolds) may lead to a combinatorial approach to 3-dimensional Poincarè Conjecture.