2015/05/31 by Matthew Stoffregen
Mathematics · #Advanced Combinatorial Mathematics #Cellular homology #Cobordism #Floer homology #Geometric and Algebraic Topology #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Khovanov homology #Mayer–Vietoris sequence #Morse homology #math.GT
paper · pdf · doi:10.1112/s0010437x19007620
published as Compositio Math. 156 (2020) 199-250 · 49 pages, 11 figures. Corrected an error in Lemma 3.8, slightly changing the statement of Theorem 1.1
arxiv created 2015/10/05 · openalex publication_date 2019/12/09 · openalex created_date 2019/12/26 · arxiv updated 2020/02/19 · openalex updated_date 2026/08/05
We compute the Pin(2) -equivariant Seiberg–Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu’s conjecture that \unicode[STIX]x1D6FD=-\unicode[STIX]x1D707 for Seifert integral homology three-spheres. We show that the Manolescu invariants \unicode[STIX]x1D6FC,\unicode[STIX]x1D6FD, and \unicode[STIX]x1D6FE give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres \unicode[STIX]x1D6F4(a1,… ,an) are not homology cobordant to any -\unicode[STIX]x1D6F4(b1,… ,bn) . We then use the same invariants to give an example of an integral homology sphere not homology cobordant to any Seifert fiber space. We also show that the Pin(2) -equivariant Seiberg–Witten Floer spectrum provides homology cobordism obstructions distinct from \unicode[STIX]x1D6FC,\unicode[STIX]x1D6FD, and \unicode[STIX]x1D6FE . In particular, we identify an \mathbbF[U] -module called connected Seiberg–Witten Floer homology, whose isomorphism class is a homology cobordism invariant.