  We prove a new non-splitting result for the cohomology of the Milnor fiber,
reminiscent of the classical result proved independently by Lazzeri, Gabrielov,
and L\^e in 1973-74.
  We do this while exploring a conjecture of Bobadilla about a stronger version
of our non-splitting result. To explore this conjecture, we define a new
numerical invariant for hypersurfaces with $1$-dimensional critical loci: the
beta invariant. The beta invariant is an invariant of the ambient
topological-type of the hypersurface, is non-negative, and is algebraically
calculable. Results about the beta invariant remove the topology from
Bobadilla's conjecture and turn it into a purely algebraic question.
