On relative simple Heffter spaces

In this paper, we introduce the concept of a relative Heffter space which simultaneously generalizes those of relative Heffter arrays and Heffter spaces. Given a subgroup J of an abelian group G, a relative Heffter space is a resolvable partial linear space whose points form a half-set of $G\backslash J$ and whose blocks are all zero-sum in G. Here we present two infinite families of relative Heffter spaces satisfying the additional condition of being simple. As a consequence, we get new results on globally simple relative Heffter arrays, on mutually orthogonal cycle decompositions and on biembeddings of cyclic cycle decompositions of the complete multipartite graph into an orientable surface.