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We analyze the space of differentiable functions on a quad-mesh $\cM$, whichare composed of 4-split spline macro-patch elements on each quadrangular face.We describe explicit transition maps across shared edges, that satisfyconditions which ensure that the space of differentiable functions is ample ona quad-mesh of arbitrary topology. These transition maps define a finitedimensional vector space of $G^{1}$ spline functions of bi-degree $\le (k,k)$on each quadrangular face of $\cM$. We determine the dimension of this space of$G^{1}$ spline functions for $k$ big enough and provide explicit constructionsof basis functions attached respectively to vertices, edges and faces. Thisconstruction requires the analysis of the module of syzygies of univariateb-spline functions with b-spline function coefficients. New results on theirgenerators and dimensions are provided. Examples of bases of $G^{1}$ splines ofsmall degree for simple topological surfaces are detailed and illustrated byparametric surface constructions.

$G 1$-smooth splines on quad meshes with 4-split macro-patch elements