On the rooted forests in triangulated closed manifolds
Sajal Kumar Mukherjee · Linear and Multilinear Algebra · 2019
In this paper, we give a combinatorial interpretation of the cardinality of the relative homology group Hd−1(F,R) for any rooted forest (F,R) in a triangulated closed d-manifold Md, which answers an open question of Olivier Bernardi and Caroline Klivans for any closed manifold. We have shown that for any rooted forest (F,R) in a closed manifold Md, the forest F simplicially collapses onto its root R if and only if Hd−1(F,R) is trivial and as a corollary we have proved that any forest in a closed triangulated orientable manifold simplicially collapses onto any of its roots.