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 $$\mathrm{xind}P\le \mathrm{xind}A+\mathrm{xind}B+1,$$ which is directly analogous to the classical union inequality for the topological index. In contrast, for every group $G\mathrm{\ne }{\mathbb{Z}}_2$, this phenomenon fails in general, and we establish the best possible weaker estimate $$\mathrm{xind}P\le \mathrm{xind}A+2(\mathrm{xind}B+1)\mathrm{.}$$ 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.