A new invariant associated with decompositions of manifolds
  We introduce a new topological invariant, which is a nonnegative integer, of compact manifolds with boundaries associated with a kind of decomposition of them. Let M and N be m-dimensional compact connected manifolds with boundaries. The new invariant of any boundary union of M and N is less than or equal to that of M (resp. that of N). 
Eiji Ogasa