@article{Xu2024, 
author = {Yan Xu and Bing Fang and Fengchun Lei},
title = {On  H′-splittings of a handlebody},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {9},
pages = {24385-24393},
keywords = {H′-splitting, Heegaard splitting, incompressible surface, primitivity, quasi-primitivity},
url = {https://www.sciopen.com/article/10.3934/math.20241187},
doi = {10.3934/math.20241187},
abstract = {Let  M be a compact connected orientable 3-manifold and  F be a compact connected orientable surface properly embedded in  M. If  F cuts  M into two handlebodies  X and  Y (i.e.,  M=X∪FY), then we say that  F is an  H′-splitting surface for  M and call  X∪FY an  H′-splitting for  M. When the  H′-splitting surface  F is incompressible in a handlebody  H, a characteristic of an  H′-splitting  H1∪FH2 to denote  H is already known. In the present paper, we generalize the above result as follows: Let  H be a handlebody of genus  g≥1,  X∪FY an  H′-splitting for  H. Then, either  X∪FY is stabilized, or there exists a reducing system  J1∪K1 of  F, such that  J1 is quasi-primitive in  Y and  K1 is quasi-primitive in  X. Combining the result with the known result, we obtain a characteristic of an  H′-splitting  H1∪FH2 to denote a handlebody.}
}