{"paper":{"title":"The geometric Hopf invariant and surgery theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.AT","authors_text":"Andrew Ranicki, Michael Crabb","submitted_at":"2016-02-29T05:52:07Z","abstract_excerpt":"The first author's geometric Hopf invariant of a stable map $F:\\Sigma^{\\infty}X \\to \\Sigma^{\\infty}Y$ is a stable ${\\mathbb Z}_2$-equivariant map $h(F):\\Sigma^{\\infty}X \\to \\Sigma^{\\infty}(Y \\wedge Y)$ constructed by an explicit difference construction applied to $(F \\wedge F)\\Delta_X - \\Delta_Y F$. The stable ${\\mathbb Z}_2$-equivariant homotopy class of $h(F)$ is the primary obstruction to desuspending $F$ up to homotopy. The explicit nature of the construction allows for a $\\pi$-equivariant version of $h(F)$ in the case of a $\\pi$-equivariant $F$, with $\\pi$ a discrete group. In earlier joi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.08832","kind":"arxiv","version":2},"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"}