On the rooted forests in triangulated closed manifolds

ABSTRACT

In this paper, we give a combinatorial interpretation of the cardinality of the relative homology group $H_{d-1}(F,R)$ for any rooted forest $(F,R)$ in a triangulated closed d-manifold $M^d$, 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 $M^d$, the forest F simplicially collapses onto its root R if and only if $H_{d-1}(F,R)$ 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.

COMMUNICATED BY:

• R. B. Bapat

Keywords:

• Simplicial complex
• cycle
• forest
• discrete morse theory
• homology

AMS Subject Classifications:

• 05C05
• 05C50
• 05E45
• 57Q10

Acknowledgements

The author would like to thank the anonymous referees for their meticulous reading of the manuscript and many valuable suggestions that significantly improved the overall exposition of this paper. The author would also like to thank one of the learned referees for suggesting the reference [8] and the remark at the end of Section 4.

Disclosure statement

No potential conflict of interest was reported by the author.