  In this paper we study the dimension of bivariate polynomial splines of mixed
smoothness on polygonal meshes. Here, "mixed smoothness" refers to the choice
of different orders of smoothness across different edges of the mesh. To study
the dimension of spaces of such splines, we use tools from Homological Algebra.
These tools were first applied to the study of splines by Billera (1988). Using
them, estimation of the spline space dimension amounts to the study of the
generalized Billera-Schenck-Stillman complex for the spline space. In
particular, when the homology in positions one and zero of this complex are
trivial, the dimension of the spline space can be computed combinatorially. We
call such spline spaces "lower-acyclic." In this paper, starting from a spline
space which is lower-acyclic, we present sufficient conditions that ensure that
the same will be true for the spline space obtained after relaxing the
smoothness requirements across a subset of the mesh edges. This general recipe
is applied in a specific setting: meshes of arbitrary topologies. We show how
our results can be used to compute the dimensions of spline spaces on
triangulations, polygonal meshes, and T-meshes with holes.
