The mapping index through the lens of the cross-index

We study the cross-index of free $G$-posets as a combinatorial analogue of the equivariant topological index. We demonstrate that the cross-index exhibits many structural properties closely paralleling those of the topological index, while its behavior with respect to unions displays a pronounced dichotomy depending on the acting group. Specifically, if $P = A \cup B$ is a union of $G$-invariant subposets, then for $G = \mathbb{Z}_2$ we obtain the sharp inequality $$\operatorname{xind} P \le \operatorname{xind} A + \operatorname{xind} B + 1,$$ which is directly analogous to the classical union inequality for the topological index. In contrast, for every group $G\neq \mathbb{Z}_2$, this phenomenon fails in general, and we establish the best possible weaker estimate $$\operatorname{xind} P \le \operatorname{xind} A + 2(\operatorname{xind} B+1).$$ This reveals a fundamental distinction between the $\mathbb{Z}_2$-equivariant and non-$\mathbb{Z}_2$-equivariant settings at the purely combinatorial level. As further consequences, we compare the cross-index with both the topological index and the simplicial index, showing in particular that the gap between the cross-index and the topological index can be arbitrarily large. These results clarify the role of the cross-index as a combinatorial analogue of the equivariant topological index and further strengthen the interplay between equivariant topological methods and combinatorial structures endowed with symmetry.