{"paper":{"title":"Deformations of box complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CO","authors_text":"Takahiro Matsushita","submitted_at":"2013-12-11T07:15:21Z","abstract_excerpt":"Box complex is a $\\mathbb{Z}_2$-space associated to a graph, and it is known that a certain $\\mathbb{Z}_2$-homotopy invariant of it, called the $\\mathbb{Z}_2$-index, gives an effective lower bound for the chromatic number. On the other hand, we show that any $\\mathbb{Z}_2$-homotopy invariant of the box complex is not equivalent to the chromatic number. Namely, we construct a graph homomorphism $f:X \\rightarrow Y$ such that it gives rise to a $\\mathbb{Z}_2$-homotopy equivalence between their box complexes, but $X$ and $Y$ have different chromatic numbers. To see this, we show that some deformat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.3051","kind":"arxiv","version":6},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}