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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.12909 |
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| _version_ | 1866916008541814784 |
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| author | Bui, Vuong Daneshpajouh, Hamid Reza Karasev, Roman |
| author_facet | Bui, Vuong Daneshpajouh, Hamid Reza Karasev, Roman |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12909 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The mapping index through the lens of the cross-index Bui, Vuong Daneshpajouh, Hamid Reza Karasev, Roman Combinatorics 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. |
| title | The mapping index through the lens of the cross-index |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.12909 |